Integral Geometry, Radon Transforms and Complex Analysis: Lectures given at the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) ... 3-12, 1996

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen

Subseries: Fondazione C. I. M. E., Firenze Advisor: Roberto Conti

1684

Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

C. A. Berenstein R E Ebenfelt S.G. Gindikin S. Helgason A.E. Tumanov

Integral Geometry, Radon Transforms and Complex Analysis Lectures given at the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Venice, Italy, June 3-12, 1996 Editors: E. Casadio Tarabusi, M. A. Picardello, G. Zampieri

Fondazione

C.I.M.E.

Springer

Authors

Editors

Carlos A. Berenstein Institute for Systems Research 221 A. V. Williams Building University of Maryland College Park, MD 20742-0001, USA

Enrico Casadio Tarabusi Dipartimento di Matematica "G. Castelnuovo" Universith di Roma "La Sapienza" Piazzale Aldo Moro, 2 00185 Roma, Italy

Peter F. Ebenfelt Department of Mathematics Royal Institute of Technology 100 44 Stockholm, Sweden Simon Gindikin Department of Mathematics Hill Center Rutgers University New Brunswick, NJ 08903-2101, USA Sigurdur Helgason Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA

Massimo A. Picardello Dipartimento di Matematica Universith di Roma "Tor Vergata" Via della Ricerca Scientifica 00133 Roma, Italy Giuseppe Zampieri Dipartimento di Matematica Pura ed Applicata Universit'~ di Padova Via Belzoni, 7 1-35131 Padova, Italy

Alexander Tumanov Department of Mathematics University of Illinois 1409 West Green Street Urbana-Champaign, IL 61801-2943, USA

Cataloging-in-Publication Data applied for Die Deutsche Bibtiothek - CIP-Einheitsaufnahme Integral geometry, radon transforms and complex analysis : held in Venezia, Italy, June 3-12. 1996 / C. A. Berenstein ... Ed.: E. Casadio Tarabusi ... - Berlin; Heidelberg; New York; Barcelona: Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1998 (Lectures given at the ...session of the Centro lnternazionale Matematico Estivo (CIME) ... ; 1996,1) (Lecture notes in mathematics; vol. 1684: Subseries; Fondazione CIME) ISBN 3-540-64207-2 Centro Internationale Matematico Estivo : Lectures given at the ... session of the Centro lnternationale Matematico Estivo (CIME) ... - Berlin; Heidelberg; New York; London; Paris; Tokyo; Hong Kong: Springer Friiher Schriftenreihe. - FriJher angezeigt u. d. T.: Centro lnternationale Matematico Estivo: Proceedings of the ... session of the Centro l nternationale Matematico Estivo (C1ME) 1996,1. Integral geometry, radon transforms and complex analysis. - 1998 Mathematics Subject Classification (1991): 43-06, 44-06, 32-06 ISSN 0075- 8434 ISBN 3-540-64207-2 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1998 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10649783 46/3143-543210 - Printed on acid-free paper

PREFACE

This book contains the notes of five short courses delivered at the Italian Canfro Internazionale Matematico Estivo (CIME) session Integral Geometry, Radon Transforms and Complex Analysis held at Ca' Dolfin in Venice (Italy) in June 1996. Three of the courses (namely those by: Carlos A. Berenstein of the University of Maryland at College Park; Sigurdur Helgason of the Massachusetts Institute of Technology; and Simon G. Gindikin of Rutgers University) dealt with various aspects of integral geometry, with a common emphasis on several kinds of Radon transforms, their properties and applications. The lectures by C. A. B e r e n s t e i n , Radon transforms, wavelets, and applications, explain the definition and properties of the classical Radon transform on the two-dimensional Euclidean space, with particular stress on localization and inversion, which can be achieved by the recent tool of wavelets. Interesting applications to Electrical Impedance Tomography (EIT) are also illustrated. The lectures by S. Helgason, Radon transforms and wave equations~ give an account of Radon transforms on Euclidean and symmetric spaces, focusing attention onto the Huygens principle and the solution of the wave equation in these environments. The lectures by S. G. Gindikin, Real integral geometry and complex analysis, give an account of the deep connection between the two main themes of this CIME session, covering several variations of the Radon transform (RT): the projective RT; RT's taken over hyperplanes of codimension higher than 1; and RT's over spheres. An important and unifying tool is the ~" operator of Gel'fand-Graev-Shapiro, used to explain analogies between inversion formulas for the various RT's. This approach goes hand-in-hand with ~-cohomotogy and hyperfunctions, typical subjects in the field of complex a~alysis. In related areas, the other two courses (namely those by: Alexander E. Tumanov of the University of Illinois at Urbana-Champaign; Peter F. Ebenfelt of the Royal Institute of Technology at Stockholm) share stress on CR manifolds and related problems. The lectures by A. E. T u m a n o v , Analytic discs and the extendibility of CR functions, provide an introduction to CR structures and deal in particular with the problem of characterizing those submanifolds of C N whose CR functions are wedge-extendible. This property turns out to be equivalent to the absence of proper submanifolds which carry the stone CR structure. (The technique of the proof consists in an infinitesimal deformation of analytic discs attached to CR submanifolds.) The lectures by P. F. E b e n f e l t , Holomorphic mappings between real analytic aubmanifolds in complex space, deal with algebralcity of locally invertible holomorphic mappings. Along with classical results, new criteria are introduced in terms of the behavior of these mappings on a real-analytic CR submanifold which is generic, minimal, and holomorphically non-degenerate in a suitable sense. To this end a fundamental tool is afforded by the so-called Segre sets.

VI We wish to express our appreciation to the authors of these notes, and to thank all the numerous participants of this CIME session for creating a lively and stimulating atmosphere. We are particularly grateful to those who contributed to the success of the session by delivering very inspiring talks. Enrico C A S A D I O T A R A B U S I Massimo A. P I C A R D E L L O Giuseppe Z A M P I E R I

TABLE OF CONTENTS

BERENSTEIN, C. A. Radon Transforms, Wavelets, and Applications Holomorphic Mappings Between Real Analytic Submanifolds in Complex Space

35

GINDIKIN, S. G.

Real Integral Geometry and Complex Analysis

70

HELGASON, S.

Radon Transforms and Wave Equations

99

TUMANOV, A. E.

Analytic Discs and The Extendibility of CR Functions

123

EBENFELT, P. F.

Radon transforms, wavelets, and applications Carlos Berenstein

We present here the informal notes of four lectures 1 given at Cs Dolfin, Venice, under the auspices of CIME. They reflect the research of the author, his collaborators, and many other people in different applications of integral geometry. This is a vast and very active area of mathematics, and we try to show it has many diverse and sometimes unexpected applications, for that reason it would impossible to be complete in the references. Nevertheless, we hope that every work relevant to these lectures, however indirectly, will either be explicitly found in the bibliography at the end or at least in the reference lists of the referenced items. I apologize in advance for any shortcomings in this respect. The audience of the lectures was composed predominantly of graduate students of universities across Italy and elsewhere in Europe, for that reason, the emphasis is not so much in rigor but in creating an understanding of the subject, good enough to be aware of its manifold applications. There are several very good general references, the most accesible to students is, in my view, [Hell. For deeper analysis of the Radon transform the reader is suggested to look in [He2] and [He3]. For a very clear explanation of the numerical algorithms of the (codimension one) Radon transform in R 2 and R 3, see [Na] and [KS]. There have also been many recent conferences on the subject of these lectures, for a glimpse into them we suggest [GG] and [GM]. Finally, I would like to thank the organizers, Enrico Casadio Tarabusi, Massimo Picardello, and Giuseppe Zampieri, for their kindness in inviting me and for the effort they exerted on the organization of this CIME session. I am also grateful to David Walnut for suggestions that improved noticeably these notes. 1. T o m o g r a p h i c i m a g i n g of s p a c e p l a s m a Space plasma is composed of electrically charged particles that are not uniformly distributed in space and are influenced by celestial bodies. The problem consists in determining the distribution function of the energy of these particles (or of their velocities) in a region of space. A typical measuring device will take discrete measurements (for instance, sample temperatures at different points in space) and then the astrophysicist will try to fit a "physically meaningful" function passing through these points. The procedure proposed in [ZCMB] is based on the idea that the measurements should directly determine the distribution function. We do it by exploiting the charged nature of the particles and using the Radon transform. (The recently launched Wind satellite carries a measuring device based on similar interaction principles and requires tomographic ideas for the processing of the data.) The advantage of the tomographic principles that we shall describe presently is that each measurement carries global information and seems to have certain noise reduction advantages over the pointwise measurements of temperatures, which is the 1These lectures reflect research of the author partially supported by the National Science Foundation.

|

](

Figure 1: Schematic detector. usual technology. We will describe everything in a two-dimensional setting, but the more realistic three-dimensional case can be handled similarly. The instrument we proposed in [ZCMB] is schematically the following. An electron enters into the instrument (a rectangular box in the figure below) through an opening located at the origin and is deflected by a constant magnetic field/~ perpendicular to the plane of the paper (see Figure 1). Under the Lorentz force, the electrons follow circular orbits and strike detectors located on the front-inside surface of the box (along the y axis). Those that strike a detector located at the point y have the property that

where m is the mass of the electron, e its charge, and B the magnitude of the magnetic field /~. In other words, all the electrons with the same first component v~ of their velocities strike the same detector located at the height y. The range of velocities over a segment of width a (width of the detector) is

Avx = (eB/2m)d (In terms of the length of the detector plate D in Figure 1 and the maximum velocity vm~ we have Avx = (d/D) - Vm~). If f(v~,vy) represents the electron velocity distribution, then the number dN of electrons counted by a detector in time dt is given by

dN = Anev~Avx i f(vx, vy)dvy, dr, --00

ne is electron density and A is the area of the entrance aperture. In other words, 1 dN f(v~, v~)dvy = AneAv~--~ --oo

so t h a t the count of hits provides the integral of f along a line vx = constant in the velocity plane. By rotating the detector or changing the orientation of the magnetic field we obtain the Radon transform of f . As a realistic example, consider a plasma of nominal electron density ne = 10 c m - 3 , velocity in the range Vmin to Vmax of 1.2 X 10 s to 3.0 • 109 cm s -1, average velocity = 6.5 • 108 cm s -1, and we assume a Gaussian distribution function

so t h a t dt

-

const, e x p , ,

2~2,]

with individual detector area and aperture of 0.04 c m 2 for a small instrument one gets t h a t the distribution function f varies from 1 to 10 -5 while d N / d t varies from 102 to 105s -1. The s t a n d a r d measurement methods make the a priori assumption t h a t f is the sum of a Gaussian centered at V and perturbed by adding a finite collection of Gaussians, often located in the region where f varies from 10 -4 to 10 -5, but the previously described instrument does not require any such assumption, on the other hand, experimentally one sees that such large variations, like from 1 to 10 -5 as in the example, are realistic. We shall see in Section 2 t h a t this is an embodiment of the Radon transform in R 2. The more realistic case of 3-d is handled by an instrument where there is a plane which contains the entrance aperture and a 2-d array of detectors in the plane (x, y). One shows t h a t at each detector location (x, y) one obtains an integral over a planar curve and that the addition of overall elements with the same x component leads to a 2-d plane integral of the density distribution so t h a t we have the Radon transform in R 3. (This is an observation we m a d e jointly with M. Shahshahani.) Before concluding this section, let us remark t h a t the large variations expected from the velocity density function f make the inversion of the Radon transform very ill-conditioned, even if f is assumed to be a smooth function. This is due to the continuity properties of the Radon transform and its inverse as seen in the next section. The remarkable point is that in medical applications, like CAT scans, the unknown density is naturally discontinuous along some curves but otherwise it has small local variations, and it is this reason the inversion problem is ultimately easier for medical applications.

Source

I []

Detector

Figure 2: Schematic CAT scanner. 2. T h e R a d o n T r a n s f o r m in R ~ Let w E S 1, w = (cos 0, sin 0), and take p E R. The equation x-w = p represents the line l which has (signed) distance p fi'om origin and is perpendicular to the direction 02.

For any reasonable function f (e.g., continuous of compact support), we can compute the line integral, with respect to Euclidean arc length ds, oo

f(x)ds = / f(x0 + tw• )dt

Rf(w,p) := / X,o2~p

(1)

--00

where x0 is a fixed point in l, i.e., satisfying the equation x0 - w = p, and w • = ( - sin 0, cos 0) is the rotate of w by ~/2. The map f ~-~ R f is called the Radon transform and R f is called the Radon transform of f. Clearly R f is a function defined on S 1 • R (that is, the family of all lines in R 2) with the obvious compatibility condition:

(R f)(-w, -p) = Rf(w, p).

(2)

There are several reasonable domains of definition for R such as LI(R~), $(R2), etc., but in many applications it is enough to consider functions which are of compact support, with singularities which are only jumps along reasonable curves, and otherwise smooth. This is obviously the transformation appearing in Section 1. The full 3-d instrument there corresponds to integration over planes in R 3, perpendicular to a unit vector w. A big source of interest of this transform lies in CAT (Computerized Axial Tomography) as a radiological tool where each planar section of a patient is scanned by X-rays as in Figure 2. In this particular case it can be seen that

I0

f

log ~ ~ / p d s Jl

(3)

where I0 is the radiation intensity at the source and I is the intensity measured at the detector. The attenuation is a consequence of traversing a tissue of density #. So the data collected from this X-ray scanning appears in the form of the Radon transform R # of the density #, computed for a finite collection of directions wt, w2,..., w~v (usually equally spaced) and a finite collection of lines, i.e., values Pl,P2,...,PM for each direction. This is called a "parallel beam" CT scanner. The configuration that it is now most used but we shall not discuss here is the "fan beam" CT scanner, we refer to [Na], [KS] for a discussion of the differences of these two cases, they really only appear at the implementation level of the inversion algorithms because only a limited amount of data can be obtained in the real world. Some easy properties of the Radon transform are obtained by observing that Rf can be written using distributions. In fact, if we introduce the unit density 5 ( p - x . w ) which is supported by the line x 9w = p, then

Rf(w, p) = f f(x)~(p - x. w)dx

(1')

R ~

with the usual abuse of language. It is also convenient to write

R~f(p) = Rf(w,p). extend Rf to (R2\{0}) x R,

using the fact that

nf(~, s) = ~ n s

,

(4)

Formula (1') can be used to is homogeneous of degree - 1 ; indeed, one defines

5(p-x.w)

(5)

One can therefore take derivatives of (1') with respect to the variables ~j(~ = (~1, ~2)) and obtain

~---~jRf(~,s)=

f f(x)~-~j~i(s-x.~)dx

(j = 1,2),

but

~

5 ( ~ - ~ . ~) = - : J ( s

- x . ~)

and ~

- x . ~) = ~ ' ( s - z . ~)

so that

0

--Rf(~, s) o~

f f(x)xjh'(s - x" ~)dx O f f(x)xjh(s- :c'~)dx 0 Os (R(xjf)(~, s)).

(6)

On the other hand the Radon transform of the derivative of f is:

= fff

fj(x)a(s-z.r

= ~j f f(x)5'(s - x . r = ~jff---~(RJ)(s).

(7)

In particular, for 02 02 A = Oz~ + Ox~' one obtains

Rr

= (~ + ~)

(RJ)(s).

(7')

When ~ is restricted to be an element co E S 2, we get, 02

( R A f )(co, s) = -~s2s2R f (co, s);

(S)

In other words, R intertwines A and ~0 2 when the arguments are restricted to S 1 • R. Another useful property is the following: P ~ ( f 9 g) = P ~ f | P~g,

(9)

where the symbol 9 on the left side of (9) denotes the convolution in R 2 and | denotes the convolution product in R. The easiest way to verify (9) is via the Fourier Slice Theorem, which we recall here: Let ~1 denote the Fourier transform of a function in R and f or F2 the Fourier transform of a function f in R 2. Then .T'l(P~f)('r) = f(Tco).

(10)

The proof is as follows, oo

Y'I(R~f)(T)

=

f e-2~it~l~f(t)dt --00

= f e-2'm"f( tco + scoX)dsdt, R 2

Letting now x = tco + sw • one has t = x- w and dtds = dx, the Lebesgue measure in R 2, in the previous equation we obtain

.T'I(/~/)(T)

= f e-2"(:"')'V(z)dz It.2

iO-~).

=

Recalling that in

R2

9F'2(f * g)(~) = ](~)~(~), we can easily prove (9). Indeed, from (10) we have

~ I ( P ~ ( f 9 g))(.)

= ~2(: 9 g)(.~) = / ( . ~ ) ~ ( . ~ ) = yl(P~/)(.)~:l(p~g)(~) = ~:l(p~f | p~g)(.)

and therefore, by the injectivity of 5vl, we get P ~ ( f 9 g) = P ~ f | Let us also note that if Ta denotes the translation by a, i.e., T , f ( x ) then R(7-,~f)(w,p) = Rif (x - a)](w,p) = P~f(p

= f(x

- a),

- w . a) = % . a R f ( p ) .

We now proceed to state some inversion formulas, which give different ways to recover f from R f . Fourier Inversion Formula: oo

(nj)O-)d~ 0

(11)

$1

The proof is clear, we begin with the Inversion Theorem for the Fourier transform. We have

f(x) = [ R2

let ( = 7-w and integrate in polar coordinates, to obtain oo

0

w6S i

We now apply the Fourier Slice Theorem to get oo

0

wES1

This inversion formula can be implemented numerically using the Fast Fourier Transform (EFT) (see [Na]). Quite often the points T w where the data J : l ( R ~ , f ) ( ~ - )

is known do not have a lattice structure. This causes problems for the F F T but we can use rebinning algorithms like [ST] to obviate this problem. To obtain another kind of inversion formula we observe the following:

fi P~f(s)g(s)ds= fi ; S(sw + tw• -oo

Let x --

sw + tw•

-oo

so that s -- x 9w, dx

-oo

= dtds, and

-r

therefore

1%2

i.e., the adjoint of P~ is the operator R~ defined by

We now consider for an

f

Rig(x) = 9(x. ~). arbitrary function g(w, s), having

the s y m m e t r y

(12) g(-w, -s) =

Rf(w,s)g(w,s)dwds= fsl dw fi R~f(s)g(w,s)ds

SlxR

-~

oo

(with the usual substitution,

x = sw + tw•

etc., we get)

= fd~ff(x)g(~,~.x)d2x $1

R 2

= /f(x)R#g(x)d2x.

(13)

l:t2

The operator R # defined by (13) is known by the name of "backprojection operator". Note, in this regard, t h a t g(w, s) is a function of "lines" and that R#g(x) is its integral over all lines passing through x. It is easy to prove the following useful property of the backprojection operator

(R#g) 9 f = R#(g | Rf),

(14)

where the convolution | in the second member clearly takes place in the second variable. This identity plays an important role in the numerical inversion of the Radon transform. Finally, we get to the following important result:

R # R f = 2 9 f.

(15)

Indeed,

R#Rf(x)

= SRf(w'w'x)dw S~

=

Sd,~zS f ( ( m - x ) m + s w • S1

--00

S~

By setting y = s~o• s =

lyl, dy = s&ods

0

we get

R#RS(x) = ~ S ~S(x § y) 0 from x, namely, let

Fz(q) := 1 fRf(~,w.x+q)dw. S1

Radon found that 0o

0

We refer to [GM] for the original 1917 paper and commentaries. An approximate implementation of (17) can be given by using the Fourier inversion formula 71-

y(x) ~ f Qo(x. ~)dO 0

where, as above, w = (cos 0, sin 0) and b

o(t) = f ITle2~t~:~l(Rf)(T)dT-b oo

"~ f I~-Ie2~'t~2:~(Rf)(T)dT" --00

(20)

1] This last approximation constitutes a band limiting process, and it can also be obtained from (14) as follows: Let wb be a "band-limiter', i.e, supp(~b) C [-b, b] and Wb = R#wb. Then (by letting g = wb in (14)), we obtain

Wb * f = R#(wb | R f) that is, we want Wb to be an approximate di-function choose Wb radial, e.g.,

(cf. [Na, Ch. 5]).

To begin with,

27r " b " where 0 < •(a) < 1, + = 0 for a > 1; this implies that wb = const. previous example is given by the ideal lowpass filter defined by

~'=

I~l+(~b ),

The

1 if0 0), so that IIfJI2 = IID~f[12 and

(D~f)'(~) = D(1/~)/(~). In other words, the group x --+ ax + b (a > O, b q R) operates via unitary operators in L2(R), and has a corresponding representation on the space of Fourier transforms (which happens to coincide with L2(R)). The "problem" of the Fourier transform representation is that the behavior of f at a point ~ depends on the values of f everywhere, for that reason, the idea of a "windowed" Fourier transform has been introduced long ago, namely, introduce a cut-off function g (say, a "smooth" approximation of X[-1,1]) and consider

9T'l((Tbg)f)(~) = f g(x -- b)f(x)c-2"i~dx. -oo

Note that 5rl((7-bg)f)(~) is r 9 f(~), where r is the wavelet r = g(x)e2~iz~,(b(x) = r If we want to consider also the behavior at f at different scales we are led naturally to the CWT: Given a wavelet r 6 L2(R), and f 6 L2(R) we define 4-

A

../+

-oo

for 0 < a < cx), b 6 R, r denotes the complex conjugate of r and denotes the L2-scalar product. We assume the wavelet is "oscillatory", that is, it is an arbitrary function in L2(R) which satisfies the condition

j

I (012 _

cr :=-oo - - - ~ a r

< oo.

17 OO

This condition implies that f r

= O. (For instance, when r is continuous at

--CO

= 0, which occurs if r E LI(R) N L2(R).) In fact, later on we will be interested in wavelets with many vanishing moments

f

xkr

= O, 0 < k < N .

--00

A typical wavelet is the Haar wavelet r = X[0,t/2] - X[t/2,t] so that

D1/2r = v~(X[0:/4]

-

X[1/4,1/2])

which shows that for k -+ co, D2-~r "analyzes" smaller and smaller details of the "signal" f. Moreover, Wcf determines f as seen from the following relation valid for any pair f,g C L2(R)

jf

dadb = cr < f,g > [1r

Wr

--00 --00

usually called Calderon's identity. If 11r property 1 Ilf-~

= 1 one also has the L2-approximation

Wcf(a, b)DaTbr

f

II ---~0

(26)

Al~lal_ 0 we have that h is n + ra - 1 times differentiable and satisfies

20 Relative exposure in a 256x256 pixels image 160

9O 80 ~, 70

19O Q

m 40

9Ol 20 10 i

L

20

40 60 80 10~ Radius of the region of interest in pixels

120

140

Figure 4: Exposure versus the radius of the ROI. (a) (b)

7jh(k)(7 ) E L~(R) N L2(R) ~ tJh(t)dt = 0

for 0 __ j _~ m, 0 < k < m + n - 1

for 0 < j _< m

--oo

Then = o(Itr ....

as ftl

and

t~+m-1II-~h C L2(R). The proof is rather elementary, it depends on the fact that if h is a function of compact support with m + 1 vanishing moments then 171~-1h(7) has n + m - 1 continuous derivatives. For ease of application it is better to work with the discrete wavelet transform (DWT). This is basically obtained by diseretizing the CWT or appealing to the multiresolution analysis of Mallat and Meyer [D], [M]. We have done this in detail in [FLBW] using coiflets [D] in order to be able to implement the inversion process using filter banks. One can show that to obtain a relative error of 0.5~o one only needs a margin of security of 12 pixels around the region of interest (ROI). For instance, to recover within this error bound an image occupying a disk of radius 20 pixels in a 256 • 256 image, one only needs about 25% of exposure, as shown in Figure 4. Figure 5 below is the Shepp-Logan phantom and its reconstruction from global fan beam data using the standard algorithm, in Figure 6 we use local data and our wavelets algorithm. The following figures are the reconstruction of a heart from real CAT scanner data using our wavelet method, and the reconstruction of the central part from local data and our wavelet method is found below.

21

(a)

(b)

Figure 5: (a) The Shepp-Logan head phantom; (b) the standard filtered backprojection in fan beam geometry (4).

Figure 6: Reconstruction from wavelet coefficients.

22

Figure 7: Reconstruction of heart from wavelet coefficients.

Figure 8: The local reconstruction of of central portion of heart.

23 We leave to the discussion and references in [FLBW] and [BW2] the comparison with other methods of inversion of the Radon transform using wavelets. One should add to the references in those two papers, the very recent work of Rubin [R], which is based on a systematic use of the Calderon reproducing formula and it is thus a development of the original ideas in [Ho].

24 4. T h e h y p e r b o l i c R a d o n t r a n s f o r m and E l e c t r i c a l I m p e d a n c e Tomography In this section we discuss the role tomography plays in a classical problem of Applied Mathematics, the inverse conductivity problem. Several of the earlier attempts to solve this problem involve generalizing the Radon transform to other geometries, that is, integrating functions over other families of curves beyond straight lines in the Euclidean plane. There are many examples of such transforms, in fact, the integration over great circles in S 2 was a transform considered by Minkowski and which inspired Radon in his work. The two we shall introduce presently are the generalized Radon transform of Beylkin [By] and the Radon transform on the hyperbolic plane [He1]. Let ~ be an open subset of R 2 and r E C ~ ( ~ • (R 2 \ (0))) be such that (a)

r

(b)

V=r

A~) ----Ar

~) for )~ > 0

r 0 for all (x,~) E ~ x (R 2 \ {0})

Then, for any s E R and w E S ~ we can define the smooth curve = {x e a : r

= s},

that is, the level curves of ~b. We let da denote the Euclidean arc length in such a curve. For u E C ~ ( ~ ) define the "Radon transform" R+u(~,s) = f

u(x)l V : r

5)ld,~(x)

/'/s ,~

Let h(x,~) be the Hessian determinant of r with respect to the second variables, 02 x h(x, ~) = d e t f ~ l then the "backprojection" operator Rr# is defined by h(x,~)

R v(x) = f

I

wES 1

Introducing K as the operator of convolution by 1/Ixl, Beylkin proved the following approximate inversion formula for the Radon transform as an operator R+: Lc2(a) --+ L~or namely,

R#cKRr = I + T

(30)

where, I is the identity map and T : L~(~) -+ L~or ) is a compact operator. In fact, Beylkin gives a recipe for a family of backprojection operators and generalized convolution operators K so that a decomposition of the type (30) holds. This gives his transform great flexibility and applicability to many problems, especially inverse acoustic problems, of course, the reader can easily verify

25 that for convenient choices of r the transform Re yields the Euclidean Radon transform studied earlier and the hyperbolic one, which we now introduce. (The reader should consult [Hell, [He3] for more details on this subject.) Let D, the unit disk of the complex plane C, be endowed with the hyperbolic metric of arc-length element ds given by 4'dzl~ ds2 - (1 -Iz[2) 2'

(31)

where Idzl denotes the Euclidean arc-length element. This metric is clearly conformal to the Euclidean metric but has constant curvature - 1 . The geodesics of this metric are the diameters of D and the segments lying in D of the Euclidean circles intersecting the unit circle COD perpendicularly. One can introduce geodesic polar coordinates z ++ (w, r), where w = z / H , r = d(z, 0). Note that Iz[ = tanh(r/2). In these coordinates the metric (31) can be rewritten as ds 2 = dr 2 + sinh 2 r dw 2 where dw 2 indicates the usual metric on cOD. The hyperbolic distance between two points is given by d(z,w)

arcsinh

Iz- !

((1-I~I~),~(I-l l )V

The Laplace-Beltrami operator AH on D can be written in terms of the Euclidean Laplacian A as

a.-

0-1zP)2a -

4 02 O 02 Or 2 + c o t h r + s i n h - ~ r o w 2.

(32)

The classical Moebius group of complex analysis is the group of orientation preserving isometrics of the hyperbolic plane D. One can define the hyperbolic Radon transform RH by Rf('~) = R H f ( 7 )

=

~ f ( z ) d s ( z ) , 7 geodesic in D

(33)

which is well defined for, say, continuous functions of compact support, or functions decaying sufficiently fast. Observe that to be integrable on the hyperbolic ray [0, oc[ (which is just the straight line segment from 0 to 1 in the complex plane C), f has to decay a bit faster than e -r. We denote by F the space of all geodesics in D, then the dual transform R # (or backprojection operator) is given by R#r

-- fr~ r

(34)

where Fz is the collection of geodesics through the point z and dttz is the normalized measure of Fz. Since a geodesic through z is determined by its starting direction w C S 1, then Fz ~ S 1 and d#z is naturally associated to ~ d w when we use this particular parameterization of Fz.

26 In order to invert R H one can proceed in the spirit of Radon's inversion formula (19). This was done by Helgason [He2, p. 155]. Or one can try to find a filtered backprojection type formula like (16). For that purpose we need to define convolution operators with respect to a radial kernel k. For k E L~oc([0, cxD)) and f E Co(D) we define f

k * f(z)

=

k

*H f(z) := ]D f(w)k(d(z, w))dm(w)

(35)

where dm(w) stands for the hyperbolic area measure, which in polar coordinates is given by

d m = sinh r drdw. Corresponding to the Euclidean formula (15) we have

R#HRHf : k * f, where k(t) -

1

Trsinh t

(36)

One can prove [BC1] that if

S(t) = cotht - 1

(37)

then

•41r

*H R

.R.

=

I

(38)

which is the exact analogue of (16). It is convenient to recall here that in the hyperbolic disk D we have a Fourier

transform [He2]. It is easier to work it out for "radial" functions as we interpret our kernel k, then the Fourier transform is defined with the help of the Legendre functions P~(r) by means of the following formula oo

k(A) = 2~rfk(t)P~_V2(cosht)sinhtdt

(A E R)

0

For radial functions k, m, we have (k * m)'(1) : k(~)~(~) So that, if/~(A) # 0 for all I E R, in principle, that is, for a convenient class of functions f, the convolution operator f , > k *H f is invertible. We refer to [He2], [BC1], [BC2], [Ku] for corresponding inversion formulas in the higher dimensional hyperbolic spaces, and the characterization of the range of the Radon transform. In particular, [Ku], [BC2] exploit the "intertwining" between RH and the Euclidean Radon transform as well as the Minkowski-Radon transform on spheres. Let us cxplain now what the above hyperbolic Radon transform has to do with Electrical Impedance Tomography (EIT) and what EIT is. Let us consider the following tomographic problem: using a collection of electrodes of the type used in electrocardiograms (EKG) uniformly distributed around the breast

27 of a patient and all lying in the same plane, introduce successively (weak) currents at each one of the electrodes (as done in EKG) and measure the induced potential at the remaining ones. The objective is to obtain an image of a cross section of the lungs to determine whether there is a collapsed lung or not. This was what Barber and Brown set up to do in 1984 [BB1], [BB2]. The point being that this equipment is cheap, transportable and provides a non-intrusive test (that is, no punctures have to be done to the chest cavity). Similarly, one can try to determine the rate of pumping of the heart using this kind of equipment. Notc that the pulse only determines the rate of contracting and expanding of the heart but not how much blood is being pumped by it. Another completely different problem arises in the determination of the existence and lengths of internal cracks in a plate, by using electrostatic measurements on the boundary [FV], [BCW], [W]. These three are examples of the following inverse problem. (The best reference for the general facts about this problem is the supply [SU]. See also the nice explanation for the general public [C], [S]): Assume/5 is a strictly positive (nice) function in the closed unit disk D. If we were to introduce a current at the boundary OD, represented by a function ~ satisfying fOBCds = 0, then the Neumann problem div (/3 g r a d u ) /3~

=0inD = r on

OD

(39)

has a solution u which is unique up to an additive constant. If r is a nice function then ~ (that is, the tangential derivative of u) is well defined on OD, so we have the input-output map 0u which is a linear continuous map from the Sobolev space H~(OD) into itself. (This statement holds for any domain D with nice boundary, not just the disk.) Consider now the (very non-linear) map /3,

>A/~

(40)

is it injective? Can one find the inverse to this map? This problem was originally posed by A. Calder6n, who proved that (40) was locally invertible near/3 = constant, more recently Nachman IN1], IN2] proved global invertibility. Since/3 is usually called the conductivity and 1//3 the impedance, this is the reason for the name EIT of this inverse problem. In the biological applications we know the value/3 for the different constituents like blood, lung tissue, etc., so one only looks for a profile of the areas occupied by them. In the determination of cracks, one can assume/3 "known", except for curves where/3 = 0, and one wants to determine this curve, or whether any exists. One can find in [SU] many important inverse problems that are equivalent to EIT: in acoustics, radiation scattering, etc. Note that in the problem of the rate of pumping of the heart, we can think that all we want to determine is just a single number, this rate. Isaacson, Newell and collaborators have in fact patented [C], [I] a device that measures this rate with the help of EIT. We also know that this problem, being an inverse elliptic problem is very ill-conditioned, so in any case one is willing to restrict oneself to find the deviation of/3 from an assumedly known conductivity/30. In the simplest case we assume/30 --= 1, so that/3 = 1 +5/3, 15/31 < < 1, and we further assume

28 5/3 = 0 on OD (One can always reduce matters to this case). Thus u = U + 5U, where U is the solution of (39) for the same boundary value, and/3 = 1. In other words AU OU

= 0 = r

in D on OD

(41)

Here A is the Euclidean Laplacian. The perturbation 5U then satisifes A(SU) o(~u) o~

= - < grad (&3), grad U > in D -- -(8/~)~b on OV

We have at our disposal the choice of inputs r

(42)

Their only restriction is that

rOD ~bd8 : O. For that reason, they can be well approximated by linear combinations

of dipoles. A dipole at a point oo c OD is given by It turns out that the solution Uw of { AU~ = 0 -r~~

o

inD on cOD

(43)

has level curves which are arcs of circles passing through w and perpendicular to 0D. That is, the level curves of U~, are exactly the geodesics of the hyperbolic metric. This fact passd unnoticed to Barber and Brown but they definitely realized that the value O(SU) (44) #Os at a point a E OD must be some sort of integral of 53 over the level curve of U~ that ends at a, precisely the geodesic starting at w and ending at a. In other words, # is a function in the space of geodesics in D considered as the hyperbolic plane, all the geodesics are obtained this way by changing w and a. Without expressly stating this, Barber and Brown introduced a "backprojection" operator that turned out to be exactly R#H and gave the approximation to ~ as R#H#. Santosa and Vogelius recognized explicitly that some sort of Radon transform was involved and used the generalized Byelkin transform and a convenient choice of K in (30) to stabilize numerically the inversion of EIT. Casadio and I, prodded by a question of Santosa and Vogelius, saw that RH was involved and developed the inversion formula (38) for this purpose. As it turns out, all of these approaches are just approximations to the linearized problem. Only in [BC3], [BC4], we realized the fact that the exact formulation of the linearized problem in terms of hyperbolic geometry requires also a convolution operator! Namely, let ~r

= c~

- 3 cosh-4(t) (45) 8r and # the boundary data (44) considered as a function on the space of geodesics in D, then one has that the exact relation between 5/? and # is given by

RH(a *H ~/~) = #

(46)

Using the backprojection operator we also obtain R# # = R# RH(g *H ~1~)

(47)

29 so that 1 G a H ( S *H (n~,,)) = ~ *H 6fl

(48)

which requires to invert the convolution operator of symbol ~. One can compute its hyperbolic Fourier transform k exactly and find out that k(.~) :fi 0 for every A E R, so that the operator ~. is, in principle, invertible, but the numerical implementation of this inversion has proven difficult so far. (Although Kuchment and his students have made in [FMLKMLPP] some progress towards implementing a numerical Fourier transform in D, which we hope will prove useful to compute 6ft.) One can recognize in (47) and (48) the same principle that lead to the numerical approach in [BB1], [SV] and others. Due to the importance of this problem there have been many other interesting approximate inversion formulas, under special assumptions on the conductivity fl, for instance, fl is "blocky", that is the linear combination with positive coefficients of a finite number of disjoint squares [DS]. Their approach is variational, and one may wonder whether one could not use some version of the Mumford-Shah edge detection algorithms [MS] to obtain a rather sharp solution of the inverse conductivity problem (40). 5. F i n a l r e m a r k The objective of these short notes (and the corresponding CIME course) was only to indicate how, beyond the well-known applications of tomography to Medicine, there are many other possible ones. Moreover, even to solve them approximately, they require deep mathematical tools, showing once more that the applicability of "pure" and "abstract" mathematics is not a fairy-tale but a concrete reality. It also indicates that it pays to "invest" one's time trying to communicate with those, be they physicists, or physicians, etc., that have the ready made applications. A lesson often lost by graduate students in Mathematics.

30 6. R e f e r e n c e s [AA] S. Andrieux and A. Ben Alda, Identification de fissures planes par une donn~e de bord unique, C.R. Acad. Sci. Paris 315 I(1992), 1323-1328. [AU] A. Aldroubi and M. Unser, editors, "Wavelets in Medicine and Biology," CRC Press, 1966, 616 pages. [BB1] D. C. Barber and B. H. Brown, Recent developments in applied potential, in "Information processing in Medical Imaging," S. Bacharach (ed.), Martinus Nijhoff, 1986, 106-121. [BB2] D. C. Barber and B. H. Brown, Progress in Electrical Impendance Tomography, in "Inverse problems in partial differential equations," D. Colton et al. (eds.), SIAM, 1990, 151-164. [BC1] C. A. Berenstein and E. Casadio Tarabusi, Inversion formulas for the kdimensional Radon transform in real hyperbolic spaces, Duke Math. J. 62 (1991), 613-632. [BC2] C. A. Berenstein and E. Casadio Tarabusi, Range of the k-dimensional Radon transform in real hyperbolic spaces, Forum Math. 5 (1993), 603-616. [BC3] C. A. Berenstein and E. Casadio Tarabusi, The inverse conductivity problem and the hyperbolic x-ray transform, in "75 years of Radon transform," S. Gindikin and P. Michor, editors, International Press, 1994, 39-44. [BC4] C. A. Berenstein and E. Casadio Tarabusi, Integral geometry in hyperbolic spaces and electrical impedance tomography, SIAM J. Appl. Math. 56 (1996), 755-764. [BCW] C. A. Berenstein, D. C. Chang and E. Wang, A nondestructive inspection method to detect a through crack by electrostatic boundary measurements, ISRTR 96-1. [BW1] C. A. Berenstein and D. Walnut, Local inversion of the Radon transform in even dimensions using wavelets, in "75 years of Radon transform," S. Gindikin and P. Michor, editors, International Press, 1994, 45-69. [BW2] C. A. Berenstein and D. Walnut, Wavelets and local tomography, in "Wavelets in Medicine and Biology," A. Aldroubi and M. Unser, editors, CRC Press, 1966. [BQ] J. Boman and E. Quinto, Support theorems for real analytic Radon transforms, Duke Math. J. 55 (1987), 943-948. [BV] K. M. Bryant and M. Vogelius, A computational algorithm to detect crack locations from electrostatic boundary measurements, Int. J. Eng. Sci. 32 (1994), 579-603. [By] G. Beylkin, The inversion problem and applications of the generalized Radon transform, Comm. Pure Appl. Math. 37 (1984), 579-599.

31 [BCR] G. Beylkin, R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math. 44 (1991), 141-183. [C] B. Cipra, Shocking images from RPI, SIAM News, July 1994, 14-15. [D] I. Daubechies, "Ten lectures on wavelets," SIAM, 1992. [DB] A.H. Delaney and Y. Bresler, Multiresolution tomographic reconstruction using wavelets, ICIP-94, 830-834. [DO] J. DeStefano and T. Olson, Wavelet localization of the Radon transform, IEEE Trans. Signal Proc. 42 (1994), 2055-2057. [DS] D. C. Dobson and F. Santosa, An image enhancement technique for electrical impedance tomography, Inverse Problems 10 (1994), 317-334. [FRK] A. Faridani, E. Ritman and K. T. Smith, Local tomography, SIAM J. Applied Math. 52 (1992), 1193-1198. [FLBW] F. Rashid-Farrokhi, K. J. R. Liu, C. A. Berenstein and D. Walnut, Waveletbased multiresolution local tomography, ISR-TR 95-73, see also ICIP-95, Washington, DC. [FLB] F. Rashid-Farrokhi, K. J. R. Liu and C. A. Berenstein, Local tomography in fan-beam geometry using wavelets, ICIP-96, Laussane. [FMP] B. Fridman, D. Ma, and V. G. Papanicolau, Solution of the linearized inverse conductivity problem in the half space, preprint Wichita St. U., 1995. [FMLKMLPP] B. Fridman, D. Ma, S. Lissianoi, P. Kuchment, M. Mogitevsky, K. Lancaster, V. Papanicolaou, and I. Ponomaryov, Numeric implementation of harmonic analysis on the hyperbolic disk, in preparation. [FV] A. Friedman and M. Vogelius, Determining cracks by boundary measurements, Indiana U. Math. J. 38 (1989), 527- 556. [GG] I. M. Gelfand and S. Gindikin, editors, "Mathematical problems of tomography," AMS, 1990. [GM] S. Gindikin and P. Michor, editors, "75 years of Radon transform," International Press, 1994. [GIN] D. Gisser, D. Isaacson, and J. Newell, Current topics in impedance imaging, Clin. Phys. Physiol. 8 (1987), 216-241. [GS] V. Guillemin and S. Sternberg, "Geometric asymptotics," AMS, 1977. [He1] S. Helgason, "The Radon transform," Birkh~user, 1980. [He2] S. Helgason, "Groups and geometric analysis," Academic Press, 1984. [He3] S. Helgason, "Geometric analysis on symmetric spaces," AMS, 1994.

32 [Ho] M. Holschneider, Inverse Radon transform through inverse wavelet transforms, Inverse Problems 7 (1991), 853-861. [J] F. John, "Plane waves and spherical means," Springer-Verlag, reprinted from originial edition Interscience, 1955. [Ka] G. Kaiser, A friendly guide to wavelets, Birkhguser, 1994. [KS] A. C. Kak and M. Slaney, "Principles of computerized tomographic imaging," IEEE Press, 1988. [KaS] P. G. Karp and F. Santosa, Non-destructive evaluation of corrosion damage using electrostatic measurements, preprint 1995. [KR] A. I. Katsevich and A. G. Ramm, New methods for finding values of jump of a function from its local tomography data, Inverse Probl. 11 (1995), 1005-1023. [Ke] F. Keinert, Inversion of k-plane transforms and applications in computer tomography, SIAM Riview 31 (1989), 273-289. [KLM] P. Kuchment, K. Lancaster and L. Mogilevskaya, On local tomography, Inverse Problems 11 (1995), 571-589. [KSh] P. Kuchment and I. Shneiberg, Some inversion formulas for SPECT, Applicable Analysis 53 (1994), 221-231. [Kul] A. Kurusa, The Radon transform on hyperbolic space, Geometriae Dedicata 40 (1991), 325-336. [Ku2] A. Kurusa, Support theorems for the totally geodesic Radon transform on constant curvature spaces, Proc. Amer. Math. Soc. 122 (1994), 429-435. [M] Y. Meyer, "Ondelettes et op&ateurs," 3 vols., Herman, 1990. [MS] J. M. Morel and S. Solimini, "Variational methods in image segmentation," Birkhguser, 1995. [N1] A. I. Nachman, Reconstruction from boundary measurements, Annals Math. 128 (1988), 531-576. IN2] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals Math. 143 (1996), 71-96. [Na] F. Natterer, "The mathematics of computerized tomography," Wiley, 1986. [O] T. Olson, Optimal time-frequency projections for localized tomography, Annals of Biomedical Engineering 23 (1995), 622-636. [Q1] E. T. Quinto, Tomographic reconstruction from incomplete data-numerical inversion of the exterior Radon transform, Inverse Problems 4 (1988), 867-876. [Q2] E. T. Quinto, Singularities of the X-ray transform and limited data tomography in l:t2 and R 3, SIAM J. Math. Anal. 24 (1993), 1215-1225.

33 [Q3] E. T. Quinto, Computed tomography and rockets, Springer Lecture Notes in Math. 1497 (1991), 261-268. [QCK] E. T. Quinto, M. Cheney, and P. Kuchment, eds., "Tomography, impedance imaging, and integral geometry," Lect. Appl. Math. 30, Amer. Math. Soc., 1994. [R]B. Rubin, Inversion and characterization of Radon transforms via continuous wavelet transforms, Hebrew Univ. TR 13, 1995/96. [RS] A. Ramm and A. I. Zaslavsky, Singularities of the Rdaon transform, Bull. Amer. Math. Soc. 25 (1993), 109-115. IS] F. Santosa, Inverse problem holds key to safe, continuous imaging, SIAM News, July 1994, 1 and 16-18. [ST] H. Schonberg and J. Timmer, The gridding method for image reconstruction by Fourier transformation, IEEE Trans. Medical Imaging 14 (1995), 596-607. [SCII] E. Sommersalo, M. Cheney, D. Isaacson, and I. Isaacson, Layer stripping: a direct numerical method for impedance imaging, Inverse Probl. 7 (1991), 899-926. [SU] J. Sylvester and G. Uhlmann, The Dirichlet to Neumann map and applications, in "Inverse problems in partial differential equations," D. Colton et al., eds., SIAM, 1990, 101-139. [SV] F. Santosa and M. Vogelius, A backprojection algorithm for electrical impedance imaging, SIAM J. Appl. Math. 50 (1990), 216-243. [W] D. Walnut, Applications of Gabor and wavelet expansions to the Radon transform, in "Probabilistic and stochastic methods in analysis," J. Byrnes et al., ed., Kluwer, 1992, 187-205. [Wa] E. Wang, Ph.D. thesis, University of Maryland, College Park, 1996. [ZCMB] Y. Zhang, M. A. Coplan, J. H. Moore and C. A. Berenstein, Computerized tomographic imaging for space plasma physics, J. Appl. Phys. 68 (1990), 58835889.

Institute for Systems Research University of Maryland College Park, MD 20742 [email protected]

HOLOMORPHIC ANALYTIC

MAPPINGS

SUBMANIFOLDS

BETWEEN IN COMPLEX

REAL SPACE

PETER EBENFELT

Department of Mathematics Royal Institute of Technology I00 44 Stockholm Sweden 1996-9-28 CONTENTS 1. Introduction 2. Preliminaries on algebraic mappings and CR geometry 2.1. Algebraic mappings 2.2 Real analytic CR submanifolds in C N 2.3 Finite type and minimality 2.4 Normal forms for generic submanifolds 2.5 The complexification of a real analytic submanifold 3. Holomorphic nondegeneracy, finite nondegeneracy, and reflection identities for holomorphic mappings 3.1. Holomorphic nondegeneracy of real analytic CR submanifolds 3.2. Finite nondegeneracy of real analytic CR submanifolds 3.3. Reflection identities for holomorphic mappings 4. The Segre sets 4.1. The Segre sets of a real analytic CR submanifold 4.2. Homogeneous submanifolds of CR dimension 1 4.3. Proof of Theorem 4.1.21 (CR dimension 1 case) 5. An application to holomorphic mappings between real algebraic submanifolds 5.1. A reformulation of Theorem 1.1 5.2. Proof of Theorem 5.1.1 6. Other applications and concluding remarks 6.1. The algebraic equivalence problem 6.2. Uniqueness of biholomorphisms between real analytic, generic submanifolds

36 1. INTRODUCTION In this paper we shall present, in fairly self-contained form, some recent ideas and concepts concerning real analytic submanifolds in C N. One of the main constructions described here is an invariant sequence of sets--called the Segre sets--attached to a real analytic submanifold in cN; this sequence of sets was introduced in joint work by the author together with Baouendi and Rothschild [BER1]. The first Segre set coincides with the so-called Segre variety, introduced by Segre [Seg] and successfully used in mapping problems for real analytic hypersurfaces by a number of authors: e.g. Webster [W1], Diederich-Webster [DW], Diederich-Fornaess [DF], Huang [H2], and others. Subsequent Segre sets turn out to be unions of Segre varieties. One of the merits of the Segre sets is that they allow one to analyze "reflection identities" (see section 3.3) for higher codimensional submanifolds--the idea of using reflection identities to analyze CR mappings goes back to e.g. Lewy ILl, Baouendi-Jacobowitz-Treves [BJT], Baouendi-Rothschild [BR1]. The Segre sets also allow a new characterization of the notion of finite type, as introduced by Bloom-Graham [BG]. In fact, the Segre sets provide a way of constructing the CR orbits of a real analytic CR submanifold without solving any differential equations (see Theorem 4.1.21). As a result, one finds e.g. that the CR orbits of a real algebraic CR submanifold are algebraic (Corollary 4.1.31). As an application and illustration of these techniques, we shall prove the following result (which we shall also reformulate in terms of more c]assical CR geometry, Theorem 5.1.1) from [BER1]. T h e o r e m 1.1 ( [ B E R 1 ] ) . Let A C C N be an irreducible real algebraic set, and Po a point in A such that Po C Ares. Suppose the following two conditions hold. (1) There is no hoIomorphic vector field (i.e. a vector field with holomorphic coe~cients and values in T I ' ~ which is tangent to an open piece of Areg. (2) If f is a germ, at a point in A, of a holomorphic ]unction in C N such that the restriction of f to A is real valued, then f is constant. Then, if H is a holomorphic map from an open neighborhood in C N of po E A into C N, with Jac H ~ O, that maps A into another real algebraic set A' with dim A' = dim A, necessarily the map H is algebraic.

The first result along these lines goes back to Poincar6 [P] who proved that a biholomorphic map H : C 2 ~-* C 2 defined near a point on a sphere S C C 2 and mapping S into another sphere S' C (;2 is rational. This was later extended to mappings between spheres in C g by Tanaka ITs]. Webster [W1] then proved that a biholomorphic mapping H : C g H C N, defined in some open subset of C N, taking a real algebraic, somewhere Levi-nondegenerate (i.e. with nondegenerate Levi form at some point) hypersurface M into another real algebraic hypersurface M ' is algebraic. Recently, Baouendi Rothschild [BR3] showed that if the real algebraic hypersurface M satisfies condition (1) of Theorem 1.1 above (which is a weaker condition than being Levi nondegenerate somewhere) then any holomorphic mapping H : C N ~-* C N, defined in some open subset of C N and with Jac H ~ 0, taking M into another real algebraic hypersurface M I is algebraic. Moreover, they show

37 that this condition is also necessary for such a conclusion to hold in the sense that if (1) is violated then there is a non-algebraic biholomorphism of M into itself. The sufficiency of condition (1) in this result by Baouendi-Rothschild is contained in Theorem 1.1 above, because a real analytic hypersurface that satisfies condition (1) automatically satisfies condition (2). We would also like to mention that the conditions (1) and (2) in Theorem 1.1 are essentially necessary for the conclusion of the theorem to hold. We refer the reader to [BER1] for details on this (see also section 6). Condition (1) was first introduced, and named holomorphic nondegeneracy (see section 3.1 for a detailed treatment of this notion), by Stanton [Stl] in connection with the study of infinitesimal CR automorphisms of real hypersurfaces. It deserves to be mentioned here that holomorphic nondegeneracy is fairly easy to verify because it turns out to be closely related to another property (finite nondegeneracy, see section 3.2), which is very computational and is a direct generalization of Levi nondegeneracy. Results of the type above for mappings between hypersurfaces in different dimensional spaces have been obtained by e.g. Webster [W2], Forstneric IF], and Huang [H1]. For higher codimensional submanifolds, work has been done by e.g. Senkin-Tumanov [TH], Tumanov [Wu2], and Sharipov-Sukhov [SS]. Other applications to rigidity properties of holomorphic mappings between real analytic submanifolds will be briefly discussed in the last section of this paper. The paper is organized as follows. In chapter 2, we give the basic definitions and facts, mostly without proofs, concerning algebraic mappings and CR geometry. More recent concepts such as holomorphic nondegeneracy and finite nondegeneracy, as well as reflection identities for holomorphic mappings, are introduced and discussed in chapter 3. The construction of the Segre sets and proofs of the main results concerning these are given in chapter 4. In chapters 5-6, applications of the techniques are discussed. A reformulation of Theorem 1.1 and a proof is given in chapter 5. Applications to uniqueness questions and some open problems are discussed in the final chapter. A c k n o w l e d g e m e n t . The author would like to thank Professors M. S. Baouendi and L. P. Rothschild for agreeing to have results and arguments from our joint papers [BER1-3] included in these notes. As the reader will no doubt notice, the results presented here, for which the author can claim any credit, are due to this above mentioned joint work. 2. PRELIMINARIES ON ALGEBRAIC MAPPINGS AND C R GEOMETRY 2.1. A l g e b r a i c m a p p i n g s . We denote by ON(po) the ring of germs of holomorphic functions in C g at p0, and by .AN (P0) the subring of ON (Po) consisting of those germs that are also algebraic, i.e. those germs for which there is a nontrivial polynomial P(Z,x) e C[Z,x] (with Z E C N and x C C) such that any representative f(Z) of the germ satisfies (2.1.1)

P(Z, f(Z)) =_O.

In particular, any function in .AN(Po) (throughout this paper we shall, without comment, identify a germ of a function with some representative of it) extends

38 as a possibly multi-valued holomorphic function in C N \ V, where V is a proper algebraic variety in C N. We list here some basic properties of algebraic holomorphic functions that will be used in the proof of Theorem 1.1. We use the notation AN for AN(O). L e m m a 2.1.2.

The following holds:

(i) If f E `AN then O~"f E r for any multi-index a. (ii) If f G `AlV and gj E ,AN with gj(O) = O, for j = 1, ..., I(, then

f(gl(Z), ...,9K(Z)) E `AN. (iii) (The Algebraic Implicit Function Theorem) Let F( Z, x) be an algebraic holomorphic function near 0 in C y x C, i.e. F E .,4N+1, and assume that

F(o, o) = o

,

OF -g-;(o, o) # o.

Then there is a unique function f ~ .AN such that x = f ( Z ) solves the equation F ( Z , x ) = O, i.e.

F(Z, f ( Z ) ) = 0 The arguments needed to prove this lemma are standard (see e.g. [BM], and also [BR3] for further properties of algebraic functions), and the proof is omitted. We say that a germ of a holomorphic mapping H : C N --~ C K at p0 is algebraic if the components of H (we write H = (H1,..., Hh')) are all algebraic. It follows from Lemma 2.1.2 (ii) that this property is invariant under algebraic changes of coordinates in C x and C K at P0 and p~ = H(p0), respectively.

2.2. R e a l a n a l y t i c C R s u b m a n i f o l d s in C N. In sections 2.2-2.5, we shall set up the notation, and give the basic definitions and results from CR geometry needed for subsequent sections. Most facts and results in these sections will be stated informally, and without proofs. Unless otherwise specified, proofs can be found in e.g. [B]. We should point out that only real analytic submanifolds will be considered. The definitions presented in these sections can be made in the broader category of smooth (e.g. C ~176submanifolds, but some of the facts stated fail to be true in that general setting. For instance, the two notions "finite type" and "minimality" presented in section 2.3 coincide for real analytic CR submanifolds, but do not coincide in general for merely smooth CR submanifolds. Let M be a real analytic submanifold in C N and P0 a point in M. Let m be the (real) codimension of M. We may describe M near P0 as the zero locus

(2.2.1)

M = {Z c cN: p(z, 2) = 0},

where p = (Pl ..... Pro) are real valued, real analytic functions near P0 with linearly independent differentials dpl, ..., dpm; we use the notation h(Z, 2) for a real analytic function in C N to indicate that we think of such objects as restrictions to ~ = Z of

39 holomorphic functions of (Z, 4) 6 C N • C N. We say that M is real algebraic if the pj can be taken to be real valued polynomials in Z and 2. The complex tangent space of M at p 6 M is defined as (2.2.2)

T;(M) = Tp(M) A Jp(TR(M))

where Jp: Tp(C N) ~ Tp(C N) denotes the complex structure in C N. The (real) dimension of Tp(M) is even and satisfies (2.2.3)

2 N -- 2m 2 such that the sections of TO(M) and their commutators of lengths < #1 evaluated at P0 span a subspace E1 of Tpo(M ) strictly bigger than E0. The multiplicity of the first Hhrmander number #1 is then gl = dim~E1 - dim~E0. Similarly, we define/*2 as the smallest integer such that the sections of of TO(M) and their commutators of lengths 2

for l = 1,2, ..., k; for k = 0, we have v 1 - 0. Similarily, we can p a r a m e t r i z e N2k by (4.1.17)

C 2kn

~ (Z, Z 1, ..., Z k - 1 , ~ 1 , "", x k ) =

A ~ (z, v2k(A)) E C N,

where (4.1.18)

v2k(A) = T k + q(z, X k, 7.k),

and recursively (4.1.19)

7.1+1 = w t + q(xZ+l, z l, w t) with w ~ = 7.z + q(z z, Xl 7.1),

for l = 1, ..., k - 1 and 7"1 = 0. Define dj to be the m a x i m a l rank of the m a p p i n g (4.1.14) or (4.1.17) (depending on whether j is odd or even) near 0 C C j ' . It is easy to see t h a t do = 0 and dl = n. In view of (4.1.10), we have do < dl _< d2 _< da _< . . . . We define the n u m b e r J0 _> 1 to be the greatest integer such t h a t we have strict inequalities (4.1.20)

do < dl < ... < djo.

Clearly, j0 is a well defined finite number because, for all j , we have dj < N = n + m and djo _> n + J0 - 1 so t h a t we have j0 _< m + 1. The dj's stabilize for j _> J0, i.e. djo = dj0+l = dj0+2 . . . . . , by the definition of the Segre sets. So far we have only considered generic submanifolds. If M is a real analytic C R submanifold of C N, then M is generic as a submanifold of its intrinsic complexification A' (see section 2.2). If M is real algebraic then A' is complex algebraic. The Segre sets of M at a point P0 E M can be defined as subsets of C N by the process described at the beginning of this subsection (i.e. by (4.1.5)) just as for generic submanifolds or they can be defined as subsets of A" by identifying X near P0 with (C~" and considering M as a generic submanifold of C ~:. It is an easy exercise (left to the reader) to show t h a t these definitions are equivalent (i.e. the l a t t e r sets are equal to the former when viewed as subsets of c N ) . The m a i n result concerning the Segre sets is the following. Let the H S r m a n d e r numbers, with multiplicity, be defined as in section 2.3.

56 T h e o r e m 4.1.21 ( B E R 1 ] ) . Let M be a real analytic CR submanifold in C N o f CR dimension n and of CR codimension rn and po E M. Assume that there are r (finite) HSrmander numbers of M at Po, counted with multiplicity. Then the following hold. (a) There is a holomorphic manifold X of (complex) dimension n + r through Po containing the maximal Segrc set Njo of M at Po (or, more precisely, every su~ciently small representative of it) such that Njo contains a relatively open subset of X . In particular, the generic dimension djo of Njo equals n Jvr,

(b) The intersection M N X is the CR orbit of the point Po in M. (c) If M is real algebraic then X is complex algebraic, i.e. X extends as an irreducible algebraic variety in C N. (d) There are holomorphic immersions Zo(to),Zl(tl) .... , Zjo(tjo ) defined near the origin,

C d~ 9 tj ~ Zj(tj) E C N,

(4.1.22)

and holomorphic maps So(tl),..., Sjo-1 (tjo), C dj ~ tj ~ sj_l(tj) E C ~i-1,

(4.1.23)

such that Zj(tj) has rank dj near the origin, Zj(tj) e Nj, and such that (Zj(tj), 2 j - l ( s j - l ( t j ) ) ) e M ,

(4.1.24)

for j = 1,...,j0. Moreover, if M is real algebraic, then all these maps are holomorphic algebraic. R e m a r k 4.1.25. The property (d) above is of technical importance as we shall see in the proof of Theorem 5.1.1 below. The mapping Zj(tj) parametrizes a piece of the Segre set Nj. Note that we do not claim that Zj(O) = 0, so the piece parametrized need not, and in general does not, contain 0. In particular, this theorem gives a new criterion for M to be of finite type (or minimal) at P0. The following is an immediate consequence of the theorem. C o r o l l a r y 4.1.26. Let M be a real analytic CR submanifold in C N of CR dimension n and of CR codimension m and Po E M. Then M is minimal at Po, if and only if ~he generic dimension djo of the maximal Segre set Njo of M at po is n + m. In particular, if M is generic, then M is minimal at Po if and only if djo = N. E x a m p l e 4.1.27. Let M C C a be the generic submanifold defined by I m w l = l z ] 2,

I m w 2 = l z ] 4.

Then M is of finite type at 0 with H6rmander numbers 2, 4. The Segre sets N1 and N2 at 0 are given by (4.1.28)

N1

= { ( Z , W l , W 2 ) : W 1 = O, W 2 =

0},

57

=

(4.1.29)

= 2izx,

= 2iz

x

, x e

C}.

Solving for X in (4.1.29) we obtain in this way (outside the plane {z = 0}) g 2 = { ( Z , W l , W 2 ) : w2 = - i w 2 1 / 2 } .

Using the definition (4.1.5), we obtain Y 3 ~- { ( Z , W l , W 2 ) : w2 = i W l ( W l / 2 - 2 z x ) ,

X E C}.

We have d3 = 3; N3 contains C 3 minus the planes {z = 0} and {wl = 0}. E x a m p l e 4.1.30. Consider M C C 3 defined by I m w l = [ z [ 2,

I m w 2 = R e w 2 [ z [ 4.

Here 2 is the only HSrmander number at the origin. Again, N1 is given by (4.1.28), and = {(z, wl,w ): z # 0,w2 = 0) u { 0 , 0 , 0 } . It is easy to see that subsequent Segre sets are equal to N2. Thus, N2 is the maximal Segre set of M at 0 , d2 = 2, and the intersection of (the closure of) N2 with M equals the CR orbit of 0. Let us also note that part (c) of Theorem 4.1.21 implies the following. C o r o l l a r y 4.1.31.

The CR orbits of a real algebraic CR submanifold are algebraic.

This corollary can be viewed as an "algebraic version" of Nagano's theorem for CR vector fields of real algebraic submanifolds. The theorem of Nagano ([N]) states that the integral manifolds of a system of vector fields, with real analytic coefficients, are real analytic. It follows that the CR orbits of a real analytic CR manifold M are real analytic submanifolds of M. However, in general the integral manifolds of a system of vector fields with real algebraic coefficients are not algebraic manifolds, as can be readily seen by examples. On the other hand, Corollary 4.1.31 implies that if, in addition, the system of vector fields comes from a CR structure embedded as a real algebraic submanifold in complex space then the integral manifolds are indeed algebraic. Before we prove Theorem 4.1.21 (in w we first discuss the homogeneous case because the proof of the theorem will essentially reduce to this case. Moreover, we shall only prove the theorem in the case where the CR dimension is 1. The general case is more technical, but the idea of the proof is the same. We refer the reader to [BER1] for the full proof of Theorem 4.1.21. 4.2. H o m o g e n e o u s s u b m a n i f o l d s o f C R d i m e n s i o n 1. Let #1 0 and Z = ( Z I , . . . , Z N ) E C N, we let /~,Z = (tvlZ1,... ,t#~Zg). A polynomial P(Z,-Z) is weighted homogeneous of degree c with ~espect to the weights # 1 , . . . , #N if P(~tZ, ~ Z ) = tcp(z, Z) for t > 0.

58 In this section, we consider submanifolds M in C N, N = n + m, of the form

wl = wl + ql(z,~)

W j -~ W j + q j ( Z, Z, Wl ,

(4.2.1)

...,//3j-1)

M: Wr=~r+qr(z,2,~t,...,ffJr--1)

Wr+l

LUr+I

Wm ~ Win,

where 0 < r < m is an integer (r = 0 corresponds to the canonically flat submanifold), and each qj, for j = 1, ..., r, is a weighted homogeneous p o l y n o m i a l of degree m j . The weight of each zj is 1 and the weight of Wk, for k = 1, ...,r, is ink. Since the defining equations of M are polynomials, we can, and we will, consider the sets No, ..., Nj0 a t t a c h e d to M at 0 as globally defined subsets of C N. Each N j is contained in an irreducible complex algebraic variety of dimension dj (here, an algebraic variety of dimension N is the whole space CN). The l a t t e r follows e.g. from the p a r a m e t r i c definitions (4.1.14) and (4.1.17) of N j and the algebraic implicit function theorem ( L e m m a 2.1.2 (iii)). We let 7rj, for j = 2, ..., m + 1, be the projection 7rj : C ~+m ~-* C n+j-1 defined by

7rj(z, Wl, ...Wm) = (Z, Wl, ..., Wj--1).

(4.2.2)

We define M j C C "+j-1 to be 7ri(M ). By the form (4.2.1) of M , it follows t h a t each M j is the C R manifold of codimension j - 1 defined by the j - 1 first equations of (4.2.1). T h r o u g h o u t this section, we work under the assumption t h a t M satifies the following. C o n d i t i o n 4.2.3.

The CR manifold M j, for j = 2, ..., r W 1, is o/finite type at O.

We shall only consider the case where the C R dimension is 1, i.e. z E C. The m a i n technical result is the following. P r o p o s i t i o n 4 . 2 . 4 ( [ B E R 1 ] ) . Let M be of the form (4.2.1) with CR dimension n = 1 and assume that M satisfies Condition 4.2.3. Let N 0 , N ] , ...,Njo be the Segre sets of M at O, and let do,dl, ...,djo be their generic dimensions. Then jo = r + 1 and dj = j , for 0 < j < r + l. Furthermore, for each j = 0 , . . . , r + l , there is a proper complex algebraic variety Vj C C j such that Nj satisfies (4.2.5)

Nj N ((C j \ Yj) x C rn-j+l) =

{(Z, Wl,

...,Win)

e ((C j \ Yj) x C m-j+l) : w k

= fjk(z,w,,

...,wj-1),

k = j, . . . , m } ,

where each fjk, for k = j , . . . , r , is a (multi-valued) algebraic function with bjk holomorphic, disjoint branches outside Vj and where f jk -- 0 for k = r + 1, ..., m. Proof. Clearly, the first statement of the proposition follows from the last one. Thus, it suffices to prove that, for each j = 0, ..., r + 1, there are algebraic functions

59

fjk and a proper algebraic variety Vj such that (4.2.5) holds. The proof of this is by induction on j. Since No = {0} and N1 = {(z,w): w = 0}, (4.2.5) holds for j = 0 and 1 with V0 = V1 = !3. We assume that there are V0, ..., Vl-1 such that (4.2.5) holds for j = 0, ..., 1 - 1. By (4.1.5), we have (4.2.6)

Nt = {(z,w): 3(x,'r) 9 *Nt-1, (Z,W,X,T) 9 .It4}.

A s s e r t i o n 4.2.7. The set of points (z, wl,...,wt-1) 9 C t such that there exists (Wl,...,Wm) 9 C m-/+l and (X,T) 9 *(N/-1 C/(V/_ 1 • c m - l + 2 ) ) with the property

that (z, w, X, T) 9 3,1 is contained in a proper algebraic variety At C C t. Proof of Assertion ~.2.7. Let S be the set of points (z, wl, ..., wt-1) 9 C t described in the assertion. Then (z, wl, ...,wt-1) 9 C t is in S if (4.2.8)

T j - ~ w j + q j ( x , Z , Wl,...,Wj_I)

,

j = 1,...,/-- 1.

for some (X, T1,..., Tl-1) 9 *(Trl(Nt_l)A (V/_ 1 X C)). (Recall the two equivalent sets of defining equations, (4.1.13), for Ad. The operation * here is taken in C t, i.e. mapping sets in C (z,wl t t ..... w,_~)to C (• We claim that the set S is contained in a proper algebraic variety At C C t. To see this, note first that (4.2.5) (which, by the induction hypothesis, holds for N , - I ) implies that ~rt(Nz-x ) is contained in a proper irreducible algebraic variety in C t. Let PI(X, rl, ..., rt-2) be a (non-trivial) polynomial that vanishes on *Vl-1 C C 1-1, and let P2(X, rl, ...,rl_l) be a (non-trivial) irreducible polynomial that vanishes on *Trt(Nt-1). Thus, if (z, wl,...,wt-1) 9 S then there exists a X 9 C such that

(4.2.9) /51(X,Z, W1, ...,Wl--2) := P l ( ~ , w 1 + ql()(,z), ..., Wl_2 + ql-2()~,Z,Wl,..., Wl-3) ) = 0 /52(x,z, wl,...,wt_l)

:= P

(x, w l +

....

= o,

i.e. R(z, Wl,... , Wl_ 1) = 0 if we denote b y / ~ the resultant of/51 and t52 as polynomials in :g. The proof will be complete (with At = / ~ - 1 ( 0 ) ) if we can show that /) is not identically 0, i.e. /51 and/52 have no common factors (it is easy to see that neither/51 nor t52 is identically 0). Note that, for arbitrary rl,..., rt-1, we have (el.

(4.M3)) (4.2.10) /52(X,Z, T1 +ql(Z,)(),...,Tt_ 1 + q t _ I ( Z , X , T1,...,Tt_2)) = P2(X, T1,...,Tt_I). It follows from this that 152 is irreducible (since P2 is irreducible). Thus,/51 and t52 cannot have any common factors because/52 itself is the only non-trivial factor of /52 and, by the form (4.2.5) of Nz-1,/52 is not independent of wt-1. This completes the proof of Assertion 4.2.7. [] We proceed with the proof of Proposition 4.2.4. Let us denote by Bt C C I-1 the proper algebraic variety with the property that (z, wl,...,wl-2) E C t-1 \ Bt implies that the polynomial/51(X, z, wl, ..., wt-2) defined by (4.2.9), considered as a polynomial in X, has the maximal number of distinct roots. Let Ct C C t denote

60 the union of Al and Bi x C. For (z, wl, ..., w~-2) fixed, let f~(z, w~,..., wl-2) C C be the domain obtained by removing from C the roots in X of the polynomial equation (4.2.11)

Pl(X,z, Wl,...,Wl_2)

=

O.

In view of Assertion 4.2.7 and the inductive hypothesis that (4.2.5) holds for Nl-1, it follows from (4.2.6) that (4.2.12) Y~ n ((Cl \ Ci) • Cr"-l+1) = {(Z, Wl,...,Wrn) e ((C j \ e l ) x

cm-j+l)

:

3 x E ~'~(Z, Wl,...,Wl--2) C C , w k -~ g t k ( X , z , w , , ...,Wk--1), ]g = l -- 1,.., m},

where (4.2.13)

g,k(X, z, Wl,... , Wk-1) = J~-l,k(X, Wl -[- (~1()(', Z), ..., Wl_ 2 + ql-2(X, z, Wl, ... , w l - 3 ) ) ...[-q k ( z , X , W l + ql()(, z), ...,Wk-1 + (~k-l(X,Z,Wx, ..-, Wk-2)),

for k = l - 1,...,m. Note that each glk, for k = I - 1,...,r, algebraic function such that all branches are holomorphic in every point (X, z, w) considered in (4.2.12), and glk -- 0 for k = Now, suppose that gm-l(X, z, Wl, ..., wz-2) actually depends (4.2.14)

c3gt l--i

is a (multi-valued) a neighborhood of r + 1, ..., m. on X, i.e.

( X ' Z ' W l ' ' ' " W l - - 2 ) ~ 0"

0 W~_l) such that one branch g of gl,t-1 is holoThen, for each ( X ~ 1 7 6 0 morphic near (X ~ z ~ w ~ ..., w~ with (4.2.15)

00gx(X0 ~ z 0 ' ~i,..., 0 0 wl-2)

r

0

and (4.2.16)

w~_, g(x ~ z ~ w ~ ..., w~_2), =

we may apply the (algebraic) implicit function theorem and deduce that there is a holomorphic branch O(z, Wl,..., Wl-1 ) of an algebraic function near (z ~ w~ w~_ 1) such that

(4.2.17)

w t - 1 - g ( O ( z , w ~ , . . . , W ~ _ l ) , Z , W l , ..., w,-2) -= 0.

Since gl,t-1 is an algebraic function, which in particular means that any two choices 0 , wl_2) 0 of branches g at (possibly different) points (X ~ z ~ wl,... can be connected via a path in (X, z, Wl,..., wt-2) space avoiding the singularities of gl,l-1 and also avoiding the zeros of Ogt,l_l/Ox, it follows that any solution 0 of (4.2.17) near a point (z ~ wl, 0 ..., wl_l) 0 can be analytically continued to any other solution near a (possibly different) point. Thus, all solutions 0 are branches of the same algebraic

61

function, and we denote that algebraic function by 01. As a consequence, there is an irreducible polynomial R t ( X , z, wl, ..., wl-1) such that X = ~t(z, wl,..., wt-1) is its root. Let Dr C C l be the zero locus of the discriminant of Ri as a polynomial in X. Outside (Ci U Dr) • C m-t+l C C re+l, we can, by solving for X = ~t(z, Wl, ..., Wl--1) in the equation

(4.2.1s)

wl-1 = gt,r-l(x, z, Wl,..., w l - 2 ) ,

describe Nr as the (multi-sheeted) graph

(4.2.19)

Wk ---- frk(Z, Wl, ...,wl-1) : : glk(Ol(Z,Wl, . . . , w l - 1 ) , W l , ..., Wk--1),

for k = l, ..., m. Clearly, we have flk = 0 for k = r + 1, ..., m. By taking V/to be the union of Cl U Di and the proper algebraic variety consisting of points where any two distinct branches of flk coincide (for some k = l, ..., m), we have completed the proof of the inductive step for j = l under the assumption that g i j _ l ( X , z , w l , ..., wl-2) actually depends on X. Now, we complete the proof of the proposition by showing that Condition 4.2.3 forces (4.2.14) to hold as long as l - 1 < r. Assume, in order to reach a contradiction, that gl,l-l(X, z, wl, ..., w~-2) does not depend on X. It is easy to verify from the form (4.2.1) of M that the sets 7rk(Nj), for j = 0, ..., k, are the Segre sets of M k at 0. Let us denote these sets by N j ( M k ) . Now, note that if we pick (z o, wl,...,Wg_~ ) o o EMt then (4.2.20)

w0 +

z0), ..., w 0

+

z 0,

...,

=

...,

Thus, if we pick the point (z ~ w~ w~_l) E M z such that it is not on the algebraic variety Ct (which is possible since the generic real submanifold M l cannot be contained in a proper algebraic variety; Cl (3 M l is a proper real algebraic subset of M t) then, by construction of Cl, the point (4.2.21)

(5 ~

W I0 • q l- ( z

-0

0 2-~-ql_2(z -0 ,Z 0 ,Wl,... 0 ,Z 0 ),...,Wl_ ,w0 3)) = (~.0,~0,...,?.~0_2)

is not in *~'1(~-1). By the induction hypothesis, rt(Nl-1) = N I - I ( M l) consists of a b l - l j - l - s h e e t e d graph (each sheet, disjoint from the other, corresponds to a branch of ft-1 j - l ) above a neighborhood of the point (z ~ w~ ..., w~ Since gl,z-1 is assumed independent of X, we can, in view of (4.2.21), take X = 2 in the defining equation

(4.2.21)

wl-1 =- gt,t-l (X, z, wl , ..., wl-2 )

for N l ( M l ) , near the point (z ~ w ~ ..., w~ From the definition (4.2.13) of gl,z-1 and (4.2.21) it follows that N I ( M ~) also consists of a b-sheeted graph, with b _< bl-l,z-1, (each sheet corresponds to a choice of branch of ~ - l , t - 1 at the point 9", l-~)) above a neighborhood of the point (z ~ w ~ ..., w~ Because of the inclusion N~_I(M t) C Nl(Mt), we must have b = bl-l,l-1 and, moreover, for

62 k! each branch ff-l,l-1 there is possibly another branch fl--l,l--1 such that for every (z, Wl,..., wl-2) the following holds

(4.2.23) f L l , , _ l ( Z , Wl, ...,w,_2) = ~k_l 1,/_l(ff. , W 1 + q I ( Z , Z), ..., Wl-- 2 "~- (~/--2(L', Z, Wl, ... , Wl--3))

+ ql_l(Z, Z, Wl + (~I(Z, Z ) , - . . , Wl-- 2 ~C q/--2(Z, Z, WI, ... , W/--3) )S i n c e all the sheets of the graphs are disjoint, the mapping k -~ k t is a permutation. We average over k and k', restrict to points (z, wl, ..., Wl-2) C M t - l , and obtain, by (4.2.21) and (4.2.23), (4.2.24)

1

bl-lfl-I

f ik- l ' t - l ( Z ' W l ' " " W l - 2 )

b,_1,_1

'

-

1 bl-ll-1

bl-lfl-1

-,

k'=l

k=l

+ q l - l ( z , 2 , ~ l , .-., @-2). Let us denote by f the holomorphic function near (z ~ Wl~ ..., w~_2) defined by (4.2.25)

1 - - I-1 f ( z , w l , ...,Wl--2) -- bl-1

bl-l,t-1

'

ft_l,t_l(Z, Wl, ...,wt-2), k

E k=l

and by K C C l the CR manifold of CR dimension 1 defined near (z ~ , w ~ "", w 1--2, ~ f ( z ~ w ~ ..., w~ by (4.2.26) g :-~ {(Z, W l , . . . , W l _ l ) : (Z, Wl, ...,Wl_2) C M l-1 , Wl-1 -~ f ( z , w l , . . . , w l - 2 ) } . The equation (4.2.24) immediately implies that K C M ~. By Condition 4.2.3, M l is of finite type near 0. Note that, by the form (4.2.1) of M, the condition that M z is of finite type at a point is only a condition on (z, Wl, ..., wz-2) (i.e. not on wt-1). Thus, by picking the point (Z o , W lo, . . . , W~_2) ~ MZ-1 sufficiently close to 0 (which is possible since, as we mentioned above, Ct M M ~ is a proper real algebraic subset of M1), we reach the desired contradiction. This completes the proof of Proposition 4.2.4. []

4.3. P r o o f of Theorem 4.1.21 ( C R dimension 1 case). By the remarks preceding the theorem, we may assume that M is generic throughout this proof. Also, as we mentioned above, we shall only prove the theorem under the assumption that the C R dimension is 1. The difficulty (which is only technical) for higher CR dimensions is in proving the equivalent of Proposition 4.2.4. We start by proving (a). Since the Segre sets of M at P0 are invaxiantly defined, we m a y choose any holomorphic coordinates near P0. Let ml < ... < mr be the H5rmander numbers of M at P0. By [BR1, Theorem 2], there are holomorphic coordinates (z, w) E C • C'* (recall that the CR dimension of M is assumed to be 1) such that the equations of M near P0 are given by wj = ff~j + q j ( z , 2 , ffJl,...,wj-1) + R j ( z , 5 , ff~) , (4.3.1)

d wk=@k + E l=r+l

hkl(z,2, w)@z

,

j = 1,...,r

k = r + l,...,rn,

63 where, for j = 1, ..., r, qj(z, 5, wl,..., t~j--1) is weighted homogeneous of degree m j, R j ( z , ~, t~) is a real analytic function whose Taylor expansion at the origin consists of terms of weight at least mj-b 1, and the hkz are real analytic functions that vanish at the origin. Here, z is assigned the weight 1, wj the weight m j for j = 1, ..., r and weight m r + 1 for j = r + 1, ..., m. Moreover, the homogeneous manifold M ~ C C N defined by wj=t~j+qj(z,2,

(4.3.2)

wk = t~k

,

t~,...,ffJj_l)

,

j=

l,...,r

k = r + l,...,d

satisfies Condition 4.2.3. For e > 0, we introduce the scaled coordinates (5, t~) E C l+m defined by (4.3.3)

wj = w j ( C v ; e ) = e ljffJj

,

j = l,...,m,

where lj = m j for j = 1, ..., r and lk = m r + 1 for k = r + 1, ..., m. We write /zkl for the function hkl( Z, Z, W; ~) --~ l hkl( Z( Z ; ~), ~'(Z;

(4.3.4)

~), 7.~(W; ~)),

and similarly,

nj(5, 5, ~; e) - ~rnj1 +1 Rj(z(5; ~), ~(5; ~), e ( e ; ~)).

(4.3.5)

Note that both hkl(5, ~, t~; e) and/~j(5, ~, t~; e) are real analytic functions of (5, tS; e) in a neighborhood of (0, 0; 0). In the scaled coordinates, M is represented by the equations

{

ff~j = (vj + qj(s 3, ~vl,..., ~vj-1) + e/)j(5, ~, t~; e)

(4.3.6)

wk=t~k+e

~ /tkl(~,5, t~;e)t~l l=rq-1

,

,

j = 1, ..., r

k=r+l,...,m,

Now, let ~J(A; e) be the mapping C jn ~-* C m, described in section 4.1, such that the Segre set N j of M at P0 is parametrized by (4.3.7)

c~ ~ h ~ (5,,~(3,; ~)) ~ c N

in the scaled coordinates ( 5 , ~ ) (cf. (4.1.14)-(4.1.16) and (4.1.17)-(4.1.19) to see how the m a p (4.3.7) is obtained from the defining equations (4.3.6)). Note that oJ depends real analytically on the small parameter e. The generic dimension dj of the Segre set Nj is the generic rank of the mapping (4.3.7) with e r 0, and is in fact independent of e. By the real analytic dependence on e there is a neighborhood lr of e = 0 such that the generic rank of (4.3.7), for all e E I\{0}, is at least the generic rank of (4.3.7) with e = 0. For e = 0 the mappings (4.3.7) parametrize the Segre

64 sets N ~ of the homogeneous manifold M ~ defined by (4.3.2). By Proposition 4.2.4, applied to the Segre sets N ~ of M ~ at 0, we deduce that the generic dimension of the maximal Segre set of M ~ at 0 is r + 1. Thus, dj0 > r + 1, where djo is the generic dimension of the maximal Segre set of M at P0 9 On the other hand, if we go back to the unscaled cordinates (z, w), we note from the construction of the Segre sets that each N j is contained in the complex manifold X = {(z, w): wr+l . . . . . w m = 0}. Thus djo < r + 1, so that we obtain the desired equality djo = r + 1. This proves part (a) of the theorem. It follows from (4.3.1) that the C R vector fields of M are all tangent to M f q X = {(z, w) e M : wj = O, j = r + 1, ..., m}. Thus, the local C R orbit of P0 is contained in M N X. Also, since there are r H6rmander numbers, the CR orbit of P0 has dimension 2 + r. Since the dimension of M 0 X is 2 + r as well, it follows that the local C R orbit of p0 is M A X. This proves part (b) of the theorem. To prove part (c) of the theorem we note that if M is real algebraic then each Segre set N j is contained in a unique irreducible complex algebraic variety of dimension dj. Since Njo contains a relatively open subset of X, this relatively open subset of X coincides with a relatively open subset of the unique algebraic variety containing Njo. Hence, X is complex algebraic. Finally, to prove part (d) one starts by defining the map Zo(to) to be the constant map with value p0. Subsequent maps are then constructed recursively from the definition of the Segre sets in normal coordinates. We omit the details of this rather simple construction and refer the reader to [BER1, Assertion 3.3.2]. This completes the proof of Theorem 4.1.21. [] 5. AN APPLICATION TO ItOLOMORPHIC MAPPINGS BETWEEN REAL ALGEBRAIC SUBMANIFOLDS 5.1. A r e f o r m u l a t i o n o f T h e o r e m 1.1. We shall reformulate Theorem 1.1 as a local result for germs of holomorphic mappings taking a real algebraic submanifold M into a real algebraic set A ~, and we shall express the conditions on M using the CR geometric notation introduced in sections 2-3. T h e o r e m 5.1.1 ( [ B E R 1 ] ) . Let M C C N be a real algebraic, connected submanifold with Po E M . Assume that M is holomorphicatly nondegenerate, generic, and that there is a point at which M is minimal. Then, if H : C N --~ C N is a germ at po of a hoIomorphic mapping, with Jac H ~ O, such that H ( M ) C A I, where A ~ C C N is a real algebraic set with dim~A I = dim~M, necessarily H is algebraic. Before we turn to the proof of Theorem 5.1.1, we show that this theorem implies Theorem 1.1. P r o p o s i t i o n 5.1.2.

Theorem 5.1.1 ~

Theorem 1.1

Proof of Proposition 5.1.2. Assume Theorem 5.1.1 has been proved. We show that Theorem 1.1 follows. Since Areg is a real algebraic submanifold, it is CR outside a proper real algebraic subset; we denote the set of points at which Areg is CR by ACR- Let M be a component of ACR such that P0 ~ M. We claim that M is holomorphicalty nondegenerate, generic, and that there is a point at which it is minimal. The algebraicity of the mapping H, and hence Theorem 1.1, follows by applying Theorem 5.1.1 to the mapping H with M defined as above.

65 First, observe that M is generic by (2) of Theorem 1.1, because if it were not then it would be contained in a proper complex algebraic submanifold of C N and hence there would be a holomorphic function vanishing on M; indeed, there would be a polynomial vanishing on it. Second, M is minimal at some point, because if it were not then all CR orbits would be proper real analytic submanifolds of M. At a point Pl E M where the dimension of the corresponding CR orbit is maximal, we may apply the Frobenius theorem and deduce that the CR orbits near that point form a local real analytic foliation of M. Thus, there is at least one real analytic, real-valued, non-constant function f on M which is constant on the CR orbits near Pl (in fact, there are l > 0, where l denotes the codimension of the CR orbit at Pl in M, such functions with linearly independent differentials). Since f is constant on the CR orbits, it is CR and, since f is also real analytic, it extends as a holomorphic function in a neighborhood of pl in C g. This violates (2) of Theorem 1.1, since f by construction is real-valued on M. Finally, M is holomorphically nondegenerate, because if it were not then there would be a point pl E M and a germ at pl of a holomorphic vector field tangent to M. It follows from Assertion 3.1.5 (by solving (3.1.6) using Cramer's rule) that then there is a holomorphic algebraic vector field X (i.e with holomorphic algebraic coefficients near Pl) tangent to M near Pl. By following this algebraic vector field into C 2N along the complexification of A,,g, we deduce that this would imply that there is a holomorphic algebraic vector field tangent to Areg at every point outside a proper real algebraic subset of Areg; we refer the reader to the proof of [BER1, Proposition 1.4.1] for the details of this argument. This contradicts (1) of Theorem 1.1. As we mentioned above, this completes the proof of Proposition 5.1.2. [] R e m a r k 5.1.3. Inspecting the proof of Proposition 5.1.2 above, one observes that the term "holomorphic" in the hypotheses (1) and (2) of Theorem 1.1 could be replaced by "holomorphic algebraic" if it were not for the fact that the Frobenius theorem (used to prove minimality of M at one point) only asserts the existence of an analytic function f (see above) and not an algebraic one. However, in [BER1] we prove an "algebraic version" of the Frobenius theorem for the CR vector fields of a real algebraic CR submanifold (by using the Segre sets); we mean algebraic version here in the same sense that Corollary 4.1.31 can be viewed as an algebraic version of the Nagano theorem. By using this result instead of the Frobenius theorem above, we can indeed replace "holomorphic" in the hypotheses (1) and (2) by "holomorphic algebraic". 5.2. P r o o f o f T h e o r e m 5.1.1. Let U C C N be a neighborhood of the point P0 to which the germ H can be extended holomorphically. We claim that we can find a point Pl E M N U such that (a) M is minimal at Pl; (b) M is ~-nondegenerate at pl, where ~ = ~(M) denotes the Levi number of the holomorphically nondegenerate submanifold M as defined following Proposition 3.2.2; (c) H is a biholomorphism at Pl.

66 To see this, recall that a real analytic CR submanifold M is minimal at Pl if and only if M is of finite type at pl. Since M is of finite type at some point, it follows from the definition of finite type that M is of finite type outside a proper real analytic subset. Also, M is f(M)-nondegenerate outside a proper real analytic subset, by Proposition 3.2.2 (iii). Finally, since Jac H 7~ 0 and M is generic (i.e. not contained in a proper complex analytic variety), Jae HIM 5~ 0 outside a proper real analytic subset. The claim above follows easily from these facts. Since H is a biholomorphism at pl, H(M) C A t, and dim~A I = d i m e M , it follows that the real algebraic set A t is a real algebraic manifold M ' at p~ = H(pl) and M t is ~-nondegenerate at p~. Thus by Proposition 3.3.1, H satisfies a reflection identity of the form (3.3.2), where the functions ~ are holomorphie algebraic near the appropriate point. For any j = 1, ...,j0 and tj E C.di near 0, we can substitute (Z, ~) = (Zj(tj), 2j-l(Sj-l(tj))), where Zj(tj) and sj-l(tj) are as defined in Theorem 4.1.21 (d), in the identity (3.3.2). We obtain

(5.2.1)

O'@H(zAtj)) = II_17

Ol~l f i

(Zj(tj), Zj_l(Sj_l(tj)), H(Zj_I(sj_I(tj))),... , - ~

_

(Zj-l(Sj-l(tj))),...),

where I~I < g + 171- Thus, if we know that

OI~IH OZC~ (Zj-l(tj-1)) is a holomorphic algebraic function of t j - l , for every multi-index (~, then it follows from (5.2.1) and Lemma 2.1.2 that

ONH is a holomorphic algebraic of tj, for every 7. Since Zo(to) is the constant map, it follows by induction that, in particular, H(Zjo (tjo)) is algebraic. Now, by Theorem 4.1.21, djo = N and since tjo ~-+Zjo (tjo) is an invertible holomorphic algebraic map, we deduce that H(Z) is algebraic. This completes the proof of Theorem 5.1.1. [] 6. OTHER APPLICATIONS AND CONCLUDING REMARKS 6.1. T h e a l g e b r a i c e q u i v a l e n c e p r o b l e m . As we remarked in the introduction, the conditions that M be (1) holomorphically nondegenerate, (2) generic, and (3) minimal somewhere are also essentially necessary for the conclusion of Theorem 5.1.1 to hold. To be more precise, if (1) or (2) is violated, then there is a nonalgebraic biholomorphism of M into itself fixing any point P0 C M. If (3) is violated and M is weighted homogeneous (with respect to P0), then there is also a nonalgebraic biholomorphisms of M into itself fixing P0; see [BER1] for proofs of these statements. However, instead of asking if all biholomorphisms taking M into M t, and P0 to p~, are algebraic one may ask the following: Suppose that there is a biholomorphism

67

H ( Z ) with H ( M ) C M ' and H(po) = P~o, is there (possibly another) one which is algebraic? This is probably true for a larger class of real algebraic submanifolds than those satisfying (1)-(3) above. Indeed, this is true for all real algebraic curves in C whereas it is not true that all conformal mappings taking one such curve into another are algebraic. (Theorem 5.1.1 does not apply, because a real algebraic curve in C is totally real and hence not minimal at any point.) As far the author knows, this problem has not been much studied, except in the cases where one can show that all maps are algebraic. 6.2. U n i q u e n e s s of b i h o l o m o r p h i s m s b e t w e e n real a n a l y t i c , g e n e r i c subm a n i f o l d s . The technique of using reflection identities combined with the Segre sets can also be used to study germs of biholomorphisms between real analytic submanifolds. Indeed, the idea of proof used to prove Theorem 5.1.1 above can also be used to prove the following uniqueness theorem. T h e o r e m 6.2.1 ( [ B E R 2 ] ) . Let M C C N be a connected, real analytic, and generic submanifold of codimension m, and let Pc G M . Assume that M is minimal at some point, and ko-nondegenerate at Pc. Then, if M I is a real analytic submanifold of codimension m and F, G are germs at pc of biholomorphisms mapping M into M I such that

(6.2.2)

OtatF

~

OlatG

(p0) = - f ~ ( p 0 ) ,

vl~[

where u is any such vector, that ((, u) # 0 (these forms are different for different u but coincide on the hyperplane).

Remark. The transition out of the first integral to the second in (3) is the simplest example of taking a real residue. We need to explain what means the substitution of the function (~, a) in the distribution 6(t) of one variable. We can do it the naive

72 way: take t = {~.x) as one of the variables, then extend it up a certain way to a complete system of variables, and apply 5(t) on the variable t. It is remarkable that the resulting form on the hyperplane will be independent of such a choice of variables. It is connected with the fact that &function has only a singularity of the first order, and we will see in the future that the situation for singularities of higher order is more complicated. Technically we can integrate in (3) on different sections of (1). Different sections give different interpretations of the projective Radon transform of which we will discuss two. 2 . T h e afflne R a d o n t r a n s f o r m . Let us take the section x0 = 1. Then we have the affine Radon transform of the function V(Y) = f ( 1 , y ) ,

y 9 IR"

Namely if ~ = ( - p , r~) then (4) IR-

where dy = dyl A . . . A dyn = w(1,y),

y)Jd ) -

(-1)~+ 1 7]j

d j, itj

if r5 r 0. Apparently the condition f E O ( - n ) gives ~o a sufficient decreasing condition (0(ly{-'~)) for the convergence of (4). Let us remind you that there is a fundamental connection between the affine Radon transform and and the Fourier transform ] :

(5)

](p,) = & _ j ( , ; ; ) .

where ~-p--0 is the one-dimensional Fourier transform. 3. T h e F u n k - M i n k o v s k l t r a n s f o r m . We obtain such a transform, if we integrate (3) on the sphera x~ + z~ + ... + x .2 = 1 . The restriction of f on the sphera will be an even function f ( x ) on the sphera and we integrate it along great spheras (sections of the sphera by hyperplanes (~, x} = 0 passing through 0). The integral on the sphera S" gives f(~) with the coefficient 2, because the sphera intersects fibers of (1) at 2 points. So we see that the affine Radon transform and the ~mk-Minkovski transform are projectively equivalent. To see the equivalence, it is essential to work with sections of line bundles (w E (.9(-n)), rather then with functions on N n and S n (in the language of functions we need to add a Jacobian). We can automatically transfer all formulas for one of these transforms to another one and do not need to develop independent theories for them. It is interesting to

73

remark that Minkovski and Funk considered their transform earlier than Radon did his. Radon knew about their results, noticed an analogy at his and Funk's inversion formulas (both used Abel's inversion formula), but missed the equivalence, between the two transforms. It is more surprising, that this tradition is alive today, and mathematicians continue to develop independent theories for these transforms. Of course sometimes the work with a smaller invariance group has advantages and the good example is the possibility to invert the Minkovski-Funk transform using the spherical functions (following Minkovski) . I believe that the projective nature of the Radon transform is one of the most important advantages it has, if compared to the almost equivalent Fourier transform. From the other side, the connection of the Radon transform with the projective structure gives the first example of the careful investigation of group invariance and geometrical structures in integral geometry. 4 . T h e affine i n v e r s i o n f o r m u l a for t h e R a d o n t r a n s f o r m . The simplest way to obtain the affine inversion formula is to combine the Fourier inversion formula OO

f(P7) e x p ( - i p ( 7 , y))pn-1 dp A aJ(7 )

f(Y) -- (n - 1)!(2~r) n 0 S~-t

with (5). Using the identity 7(p~ -1) = (~ - i)!i~ (p- io)-" = (~ - i)!i" p -~ -~(-i)" ~(~-~)(p) we obtain OO

(6)

f(Y) =

~

/(7,P)(P - (7, Y) - iO)-ndP A w(T). --00

Sn--1

It is essential that the integrand can be pushed down on the sphera S n-1 rather then on p n - 1 (more precisely, we can integrate it on S n-1 or any other surface F, which intersects almost any ray out of 0 once). Of course we can leave only the even part in the integrand. So we need to take only the even part of (p - i0) - n which coincides with the first term for even n, and the second term for odd n. As the result we have the local inversion formula for odd n and nonlocal for even n. In both cases we can integrate on p n - 1 , but at (6), which is true for all n, we integrate on the sphera. The reason is that ( p - i 0 ) - n is only positive homogeneous. We will interpret the formula (6) the following way. Let us consider the modified Radon transform

(7)

?(7,~0) = c

i](7,p). ---r-~

p t t0

ap,

I m ~0 > 0,

~ = ((0,7).

--oO

It will be holomorphic of (0 on the upper half-plane, C+ = {z : Im z > 0} and it will not be even. Let

(s)

~](zl~, ~7) = c?~: -1)((7, ~), 77)~(7)

74 where

f~(,,-1)_ Of~('*-:) ~0

(O~0p-1

We have the (n - 1)-form on sphera S n-1 which also depends holomorphically on complex parameters z out of the half-spaces C~ = = {z e C ~ : Im(~,z} > 0}. In a general situation, we can not integrate this form on S n-1 because the halfspaces Cg have no joint points for 77 E S '~-1. If tr has boundary values of some sort on R ~, then we can integrate n ] and we will have

(9)

f(x) = c /

n](xl~,d~),

x e ~n

S,-1 For f E O(-n) boundary values exist at least in the $1-sense. Let ](77; p) = 0 if + r / ~ cl(V) and ](r]; p), ~ e If, admits a holomorphic extension on the lower half-plane C_ on p , where V is a sharp (does not contained lines) convex open cone (it is equivalent to the condition supp ] C cl(Y)). Then ](r/; p) for -r] E V is holomorphic at C+ and

?(r/;p) = ](r/;p),

, e v,

? ( , ; p ) - o,

, ~ el(v)

For such f we can extend (9) into the complex domain:

(9')

f(z) = / nf(z[7], &l)

has sense for z E T* = 11~n + iV* where

V* = {y E An: (y,t) > 0 for allt C cl(V)\(O)) is the dual cone to V and we obtain a holomorphic function at the tube T*. The integral on the sphera at (9 ~) reduces to the integral on V N S ('~-1). Conversely holomorphic functions at the tube T* for different functional spaces of Hardy's type-H2(T*), $(T*), ,.q'(T*) etc. (Laplace-dual spaces to L2(V), $(Y), $'(Y) etc.) have the holomorphic (on p) Radon transform satisfying to the above conditions. It is possible to interpret the representation (9 I) as Radon's analog of the Fourier-Laplace transform. 5. R a d o n ' s r e p r e s e n t a t i o n o f h y p e r f u n c t i o n s . In the general situation (9) gives a realization of f as a hyperfunction - cohomology class from H ( n - 1 ) ( C n \ R n, (9). Let us give a realization of this class in the Cech language. If we have a covering of Cn\llU ~ by (n + 1) half-spaces

C~={z;Im(r/j,x}},

j = 1,...,n+1,

75 and Tj* are the intersections Nk#j C~; the intersection of all C ] is empty. Then (n - 1)-cocycles for this cohomology are the collections of holomorphic functions

fj e O(T;)

{fl,. '' ,fn+l}, without any conditions or identifications. elements of H(n-1)(C~\nt ~, O). If

So such collections are precisely the

Tj* = N" +iVj* then it is simple to see that the dual cones Vj are mutually disjoint and the union of their closures gives whole N n. Let

?j Following up on our remark, it will be the Radon transform of a holomorphic function fj at the tube Tj. If we put functions f in the correspondence with the collections { f l , . . . , f~+l }, then we obtain the realization of functions f as the ((~ech) cohomology classes { f l , . . . , f n + l } e H('*-l)(Cn\Nn,O),fj E O(T;). If fj(x), x E R n, are boundary values of fj then

f(x) = f l ( x ) + . . . +

fn+l(X),

x e ~n

We used in this construction the existence of finite coverings of C n \ R " by the Stein manifolds C~. The existence of a finite covering by Stein manifolds for a complex manifold is a very rare phenomena. Let us remark also that the finite covering in our example is noninvariant relative to linear authomorphisms. The Radon transform gives a hint that at cases of infinite coverings it is natural to work with a continuous version of Cech cohomology [G1],[G2] (cf. other examples at Lecture 2). Namely we have the covering of C n \ N '~ by half-spaces C~ = {z 6 C", Im 0},

~ 6 S "-1

So S n-1 is the parametric manifold. Let us consider differential (n - 1)-forms

dr/) which holomorphically depend on z E C~ as of parameters. So parametrical domains differ for different r/E S n-1. Let us consider the complex of such forms with parameters. In our case forms w have maximal degree and are automatically closed so we need only to factorize their space of all forms on the subspace of exact forms. It is possible to prove that this quotient is isomorphic to H ( n - 1 ) ( C n \ R ~, (9). It is the consequence of a very general result about the continuous Cech cohomology (it is important only that the submanifold Sz of parameters r/corresponding to the Stein domains C~ containing a fixed point z is contractible). The Radon transform gives a very special representation of f E O ( - n ) as a form aJ(z]r/, dr/). Namely, the form ~;f(z]rh drl) as a function of z depends only on (r/, z}. on S n-1

76 It is possible to interpret the Radon transform as a tool to obtain such special representations of hyperfunctions. If f E S, then such a representation ,; does exist, and is unique if it is put on w(z] .) as a function of z the natural S-conditions of smoothness and decreasing and moreover there are no exact forms with such conditions. The same is the situation with f E L 2 either f E C' or other space of decreasing distributions. We observe another situation for f E S ' : nontrivial exact forms w with S~-restrictions exist and they correspond to nonessential distributions in the sense of Gelfand - Graev [GGV]. It is usually the case that the description of the Radon image of spaces of increasing distributions must include a factorization and a cohomological language is very natural for this description. It would be interesting to investigate the general problem of "Radon" representations of cohomology-hyperfunctions. There is a very simple explicit (chain) morphism out of continuous Cech cohomology to the corresponding Dolbeault cohomology. Namely let y = 7(z) be any function (section) o n c n \ ~ n such that

Im(~(z),z) > 0. Then

(~o)

~ ~(zl~(z), d~(z))] ( ~

induces a morphism out of the continuous cohomology to the Dolbeault cohomology. At (10) we restrict w on a section ~ -- 7(z) and then take (0, n - 1)-part of the resulting form on c n \ R n. 6. P r o j e c t i v e i n v e r s i o n f o r m u l a . We want to give a projective invaxiant version of the inversion formula. The first step is to replace our affine inversion formula so that the formula will depend on any plane on infinity ((,z) = 0

z EPn

and our affine inversion formula (9) will correspond to ~ = (1, 0 , . . . , 0). It is simple to rewrite formulas. Let

(11)

T(z] + A~]~) ----c

(12)

~](zl~,d~lr

/ ](7p - -+~Pr dp,

q, ~ E P n , I m A > 0.

1 ( d ) n-1 = (r

f~(~ + Ar162162

,d,],

Here z is a point of the affine semispace T,,r

(13)

(r/, z)

T,,r = {z E CPn; Im ~

< 0}.

(~,z) A = - - -(r z)'

77 If F(() is any cycle intersecting any ray out of ( once (e.g. sphera Srn--1 with the center ~), then 2r~,r and ~ ] depend only on a ray through r / a n d the family of is a covering of (14)

Cn(~)\R"(~),

C " ( ( ) = C P n \ { ( ~ , x ) = O}

R"(()=Cn(~)

Tn,r

N RP"

So ~ ] gives a cohomology class (on the language of continuous Cech cohomology) at C n ( ( ) \ Rn((). If x ] has boundary values on z = x E lI~n(() then we have an inversion formula f

(15)

f(~) = c [

~](xl,,d,),

x ~ R"(().

r(~) We obtain the affme formula (9) for ( = ( 1 , 0 , . . . , 0). Of course this construction is already projectively invariant, but it is much more interesting to find a formula where the affmization ~ is variable (depends of r/). All forms in formulas (12) represent the same functional for different ~, but they do not coincide for different (: they differ on exact forms. The idea is to construct a closed form ~ ] so that it includes differentials d( and for fixed ~ coincides with (12). It is possible to compute such an extension using only the closure of the form, but we will do it using another way. So our goal is to give an invariant sense to the functional

(16)

f/(~)( l ; 6~k ) = l , ~}k)=O, i#k. We have the cycle of all q-planes in the fixed (q + 1)-plane j > 1}

{y : yq+j -= p(J),

(the Euler cycle) and the restriction of a~ on this cycle will coincide with the form in the Radon inversion formula for this (q + 1)-plane: Oq c--@(70),~ 0(p(1))q

(q+2) .,~(n);p(1),yq+2,...,yn)lpO)=(r ''"

[ (1) d (1) , (1) ] x [r/[q+ll , r/[q+ll,...,a7[q+l] ]

7Is] = ( 7 1 , . . . , 7 s , 0 , . . .

) ,0).

The next example is a little bit more complicated: n = nl + n2, q = ql + q2, qj < nj,

rj = nj - qj;

q(j) = ~(q2+J),

rl(J) = ~(ql+j),

1 < j _< nl;

nl+l<j.

The inversion formula for this cycle reduces to the pair of the Radon inversion formulas in the pair of transversal planes of dimensions ql + 1 and q2 + 1. This cycle 3' for n = 4, q = 2, nl = n2 = 2, ql = q2 = 1 is not homological to 0, ~93 is not exact (q = 2 is even), but c(7) is 0. Remarks. 1. The closure of the form t@ is the consequence (and moreover equivalent) to the system of differential equations

(6)

o 2]

(i)

(k)

0(j 0(,

(i)

(k) --

0( t 0~

o

83 It is simple to obtain (6) by the direct differentiation of (1) (but it is not simple to derive the closure of g~3 out of (6)). We will call (6) by the equations of F.John (who defined them in the case n = 3, q = 1).It is possible to prove that the space of the sections F ( ~ I , . . . , ~ ) 6 O ( - 1 , . . . , - 1 ) satisfying to (6), is exactly the image of O ( - q - 1)) on P " for the Radon-John transform (we consider C r162sections). For r = 1 the system (6) is trivial. It is the Paley-Wiener theorem for the projective R a d o n - J o h n transform. The affine version of this theorem image of S(•"). If q > 1 and we put the precise definition), then the John If q = 1 then (6) is empty, but there [GGV]): momentums

is more complicated. Let us investigate the the natural S-conditions on r (we drop system (6) gives the description of the image. are the Cavalieri conditions (Gelfand-Graev,

xk~(~) = j ~(,7;p)pk@ --00

are polynomials on 7? of degree k. These conditions are equivalent to the smoothness ~(~) at 0. For q > 1 it is also possible to formulate the analog of the Cavalieri conditions, however these integral conditions are the consequence of the differential conditions (6). This remarkable theorem was proved by Gelfand-Graev-Shapiro

[CGS]. 2. In the integral geometry we usually consider the complexes r (n-dimensional submanifolds) of q-planes and we want to reconstruct ~o only through c~li-. If 7 = Fv is the submanifold of planes from P passing through y, then for generic y EII{n we can expect that dim 7 = q and if it is the cycle we can use (5) to reconstruct ~(y). The most serious obstruction is that as a rule we can not compute the result of application of the differential operator ~ to qo for r] 6 7 only through the restriction of ~ on F. If such computations are possible for all y the complex r is called admissible. It is a very degenerate condition. In a sense P must be characteristic

for (6). The simplest class of admissible complexes is the class of translation-invariant complexes. So we can start of any q-cycle 7 of q-planes through a point y and take the complex P of all q-planes parallel to the planes out of 7. These complexes are admissible and if c(7) # 0, then we can reconstruct ~ through ~31r.

The admissible complexes of lines are classified, however admissible complexes for q > 1 are not classified and moreover it is not easy to construct explicit examples of such complexes. Let us mention only one class of examples [G4]. Let A be a q-cycle in the Grassmanian Gr(q, n + 1) (the manifold of (q - 1)-planes at p n ) and F(A) be the complex of q-planes which contain at least one of planes of A. Then the complex F(A) is admissible. The special case of such complexes for q = 1 is well known: complexes of lines intersecting a curve. Admissible complexes are very rare and are a degenerate class of complexes, but they appear in some important applications: representations theory, nonlinear differential equations.

84 In dual parameters operator n has the following form:

(7) kT(x;ul,...,uq,du('),...,du

(q))= A \ou(O)

o +...+,:,

5> 0

=

~_, I=(il,...,iq)

(o) r

o ~ ( ~ ) (o) d~!~ . ~, ,, h . . . A a u q h,(o)=x,

., ( o ) - - ' - ~ O'Uil

n i ..... /

x E

~tn.

' " " UUiq

We differentiate here on the coordinates of u ~ and take differentials of the corresponding coordinates of other vectors u i in the special order. As the result this form of dr/(1),..., dr~(q) which depends on parameters x E R n, is a combination of differentia/monomials of the very special form. Such combinations can be characterized by the following condition: they are zero on any submanifold of planes containing a fixed point. In such a form the operator ~ was introduced by GelfandGraev-Shapiro. This representation is projective. It is possible to derive this formula out of (4) but it is simpler to investigate this operator independently. By the direct differentiation of (2) we obtain that

02~ (8)

~

(0~

02~

(k)

ouj ou I

(k)_~ (0

~

r

= 0

uu!

and then again directly we can see that

d(~)

= o

so ~ is closed (we take the differentials only on r/(J), j > 0). We can check the analog of (5) for the base cycles. Usually this version of ~ is more convenient for computations. 3. N o n l o c a l i n v e r s i o n f o r m u l a s a n d O- c o h o m o l o g y in n o n c o n v e x t u b e s . Ir~ Lecture 1 for r = 1 we constructed the universal inversion formula using (p - i0) - n which works in both local and nonlocal cases. Unfortunately this construction does not generalize for r > 1 at the full volume. To understand obstructions in the way we will interpret the operator ~ in the language of densities. Let us compute the evaluation of the form ~ ( p , r/; dr/) on some r-dimensional tangent subspace at the point r/0. It is convenient to identify locally this subspace with the r-plane L passing through r/0 (at r/-coordinates). If dT1,..., dvr are some coordinates on this subspace and ~ = pr/where r/E L, then at r/0:

d~ = ,](p) dp dr The Jaeobiaa J is the polynomial on p which depends on 7/0, L and J ( p ) is the symbol of the differential operator which gives ~ on L:

(9)

0

~ ( y , r/0, dr/)lL = Y(~)~(r/0;

P)]p:"

The trouble is that for the reproduction of the integration on ~ we need to integrate IJ(P)] rather than J ( p ) . When q is even, it can be seen to occur that the

85 J a c o b i a n J(p) conservs the sign along a cycle r/E 7 and then by untegration of ~q3 on 7 we will inverse the R a d o n - J o h n transform. In the general case we can replace the differential form ~3 by a n o t h e r analytic object - the even q-density A~ whose evaluation on q-vectors will use the pseudodifferential o p e r a t o r with the symbol [fl(p)[ instead of the differential o p e r a t o r with the symbol ,](p). The q-density A~3 which in the contrast to the differential form ~3 will not be a multilinear form on dr/. The inversion formulas using the q-densities were considered in [GG]. In Lecture 1 for r = 1 we replaced the integration along lines with the integration along rays where the Jacobians have constant signs and and as the result we worked with the differential forms on the manifolds of rays S (~-1). This trick does not works for r > 1 for m a n y reasons: we can not divide r-planes on non trivial convex cones, a n d also the domains where Jacobians have constant sign d e p e n d on L etc. However there is a special situation when it is possible to realize this plan for r > 1 a n d it brings interesting consequences. Let V be an open cone in R n. It is essential t h a t we do not suppose t h a t V is convex. L a t e r we will p u t some strong restrictions on V. You can be guided by the following 3 examples: 1. T h e qudratic cones V(q+l,r-1)

= { ~ e l ~ n ; ~ 2 + . . ' + ~ q2+ l - ~ q +2~ . . . . .

~n2 > 0},

r + q = n, q > 0.

If q = 0 then V(1, n - 1) has 2 connected convex components: the future and past light cones. 2. V = M+(m) consists of all matrixes of the order m with the positive determinant. 3. V = Sym(~, m - s) consists of all symmetric matrixes of the order m with the signature (8, m - s). For s = m or s -- 0 this cone is convex. Let us consider a space of functions or distributions r with s u p p o r t s in V (e.g. r E L2(V), or S ' ( V ) ) . The problem is to describe the Fourier-dual spaces. If the cone V is convex, then there are known P a l e y - W i e n e r - B o c h n e r theorems of different kinds. T h e y give the descriptions of the corresponding H a r d y t y p e spaces of holomorphic functions in tubes T* = R n + iV* (cf. Lecture 1) a n d the F o u r i e r - L a p l a c e transform establishes the isomorphisms between the two series of the spaces. Roughly speaking H2(T), $'(T) etc. consist of holomorphic functions in the tubes T which have the b o u n d a r y values on R n in the topology of the corresponding spaces L 2, $ ' etc ( a b o u t H2(T) cf. below). We will describe below some analogs of these results for some nonconvex cones [Gb]. For the definiteness we will talk about L2(V) and later we will make some remarks a b o u t S'(V). Let us consider some geometrical constructions for nonconvex cones. If the intersection of V with an r-subspace L has the connected component V(L) which is a convex sharp cone in this plane, then we will call V(L) by the r-slice of V. Of course one subspace L can contain several slices. A c t u a l l y we are interested in r-slices for such a m a x i m a l r t h a t the union of the r-slices coincides with V. Moreover often we do not consider all slices, b u t only a set sufficient for a covering of V. In E x a m p l e 1 we have r-slices - 2 components of sections be the subspace {~2 . . . . . (q+l = 0} or more general V(w) = {~; ~+ =

86

Aw, A > 0}, where ~o E Sq and (+ = (~1,... ,~q+l, 0,..., 0). This family of slices is a l r e a d y sufficient for the covering, but we can extend it by the action of the group SO(q + 1, r - 1) of the a u t h o m o r p h i s m s of the cone. There are no/-slices for l > r. In E x a m p l e 2 the cone V(e) = Sym(m, 0) of symmetric positive m a t r i x e s is the r-slice, r = m(m + 1)/2. We will o b t a i n the slices Y(gl, g2) if transform Y(e) by the a u t h o m o r p h i s m s of V:

~ gl(g2,

gl, g2 E GL(m; N)

In E x a m p l e 3 let us consider the subcone V(e) of the matrixes

(A

BO) '

A E Sym(l, O), B E Sym(O, m - l).

Here A are symmetric positive matrixes of the order 1 and B are symmetric negative matrixes of the order r n - l . It is the r-slice with r = l(l+l)/2 + ( m - l + l ) ( r n - l ) / 2 as well as its translations V(g) by the a u t h o m o r p h i s m s of V

~ g~gT,

g E GL(m; R).

For general cones we will parameterize the r-slices in the following way. Let H~(V) = I I ( Y ) be the manifold of such r-frames r] = {~J} t h a t the positive r - e d r o n

{~ = P~;Pl > O,...,pr > 0 } contains one and only one r-slice V(q). Then to each 7] E I I ( V ) there corresponds only one slice V (but different r / c a n give the same slice V(r/). T h e image of V(rl) at the coordinate space N~ relative to the base r1 will be a cone v(rl) inside the positive r-edron. Let (10)

~(r]; p) = ~-p--p~2(P~)ipev(,)"

This function as the function of p E R ~ will be extendable as a holomorphic function at the t u b e t*(~) = W + i v * ( , ) C C ~, where v(r/)* is the cone dual to v(rl). Let us now (cf.(4)) put

(11)

= c[,(1),.

( )Iql] ?(,;

This form as a function of the p a r a m e t e r s z will be holomorphic in the convex t u b e domain (12)

T*(q) =I~ n +iV*(~),

87 where V*(r}) is the dual cone to the convex (but non open) cone V(r/) C R n. Since the cone V(r/) is contained in the r-subspace {r}} the cone V*(r/) will be invariant relatively to all translations in the direction of the orthogonal subspace {r/}Z(the cone V*(r/) will not be sharp). We have (13)

Y*(,) = {y E R"; (,, y) E v*(,)}

In (11) we obtain objects which we considered in Lecture 1 - the differential forms with holomorphic parameters. It is natural to connect them with q-dimensional O-cohomology of the nonconvex tube (14)

T* =

n + iV*,

where V* = U v * ( , ) . yen

We can consider V* as the analog of the dual cone for nonconvex cones. Let us call cones V*(r/) by the wedges of V*. We obtain the operator to the cohomology in quite a general situation, but without strong conditions on the geometry of the cone V the induced operator on the cohomology can have a big kernel (or sometimes can be trivial) and we do not have the correspondence between functions on R n and cohomology. It is not a surprise that we can not work with arbitrary nonconvex cones. Now it is the time to define a class of "good" noconvex cones.

D e f i n i t i o n . The cone V is called regular with the convexity index r (the concavity index q -- n - r) if there is such a q- dimensional cycle 7 C II(r) that (i) = v. (ii) The slices V(r/), 77 E 7, are mutually disjoint. (iii) For any 3 slices Y(~), Y(r/), V((); ~, r/, r C 7, the slice V(r either is contained in the convex envelope Y(~) + Y(r/) of Y(~) and Y(r/) or does not intersect with it (concavity condition).

Under these conditions we will call the cycle 7 as the concave generating cycle. We will work below only with regular cones V and concave generating cycles 7. In the definition of V* we will replace II on 7. In the regular case r-slices V(r/) are sharp convex cones of the maximal dimension which are component of plane sections of Y. Their dual V*(r}) (wedges) are the maximal convex subcones of V*. The condition (ii) means that the convex envelope of any 2 wedges is the whole space R n. The condition (iii) is the dual of the condition that any wedge can either contain the intersection of 2 other wedges or the convex envelope of it and the intersection is the whole space. It would be interesting to develop the geometry of the regular cones. Several problems at first: To prove that the construction of V* where in (14) we take a generating cycle 9' instead H is independent of the choice of the cycle 7 and coincides with the construction (14). I believe that it follows out of some analytical results on hyperfunctions and wave fronts, but it would be interesting to give a direct geometrical proof. Another question: is the cone V* regular if V is regular? If the answer is positive then is it true, that (V*)* = V? Let us turn to our examples:

88

1. For the cone V(q + 1, r - 1) the family of r-slices V(w) will be the cycle 7,2_ S q satisfying all conditions of the concave generating cycle. The dual cones V*(w) - wedges of V * - are obtained out of the slices by the direct product with the orthogonal q-subspaces. It is remarkable that their union will again give the cone V(q + 1, r - 1). So this cone is sel~dual relative to the defined duality: V = V*. 2. For V = M + ( m ) we parameterize points of the cycle 7 by the elements u of the orthogonal group S O ( m ) and take the slices V ( e , u) The verification of (i) and (ii) is trivial, of (iii) is the exercise in linear algebra. If to use the duality on the matrix space M ( m ) by the bilinear form

(r v) = tr (r y), then V*(e) = {x 9 M ( m ) ; (x + x c) 9 Sym(m, 0)}. So this cone is the direct product of the slice V(e) on the subspace of skewsymmetric matrixes, (q = m ( m - 1)/2). It is a good exercise to prove that this cone will be inside M +. Correspondingly V*(u) = u. V*(e) and we obtain that the cone M + is self-dual. 3. For the cone in Example 3 slices V ( u ) , u 9 S O ( m ) , give the concave generating cycle 7. We have V ( g ) = V(e) if g =

(0" 0),

A'9

K

Therefore the cycle is isomorphic to S O ( m ) / S O ( 1 ) • S O ( m - l) and dim 7 = q = t • (m - l). This cone is also self-dual. We had considered 3 classes of self-dual homogeneous cones; about such cones refer to [FG]. Let us mention that the cone V = R n \ {0} is a self-dual regular cone. The rays are its 1-slices V(r/), r / 9 S "-~, the half-spaces are the dual wedges V*(r/) and the only regular cycle is the sphera S n-1. In this way the results in the end of Lecture 1 are the partial case of the results of this Lecture. For any generating cycle 7 we have

(15)

f ~ ( x , 7, dr/) = c(~)~(x),

~ e L~(V),

x 9 ~tn,

c(~) # 0.

3~

We take here the boundary values of n ~ ( z , r/, dr/) in L2-sense. We need only to remark that by virtue of the condition (ii) of the definition of regular cones the Jacobians f f will have no zeroes on V(r/),7? E 7, and it means as we already remarked, that the integral at (15) gives the integral on l~n at the Fourier inversion formula. Thus we connected with ip(x) the form representing a continuous Cech cohomology class in the tube T* relative to the covering {T*(r/), r/E 7}. For this manifold we can not avoid the use of the infinite Stein coverings (depending on continuous parameters). Let us look on the usual Cech cochains for our covering. Suppose for

89

the simplicity that q = 1. Using the concavity (iii) of the cycle 3' we can define an ordering on the ~,: -~~77 if V ( ( ) C ( V ( ~ ) M V ( r / ) ) Put

j([~

a ~ ( z ; r de),

r

') =

z 9 T*(() fl T*(r/).

,,1

By virtue of (iii) we have a 1-cochain and moreover r

•) = r

() + r

U)

onT*(~) M T*(rl) = T*(~) a T*(q) M T*(~)

It shows that the system of r ~) is extremely overdeterminate and the transition to continuous cohomology is a way to rationalize this information. It is convenient sometimes to use some quasipolyedron approximations. Let X = { t / ( i b . . . , q(N)} be such a cyclic sequence of points in 7, that between points with sequential numbers are no other points of AF. Let us put

VN =

U

V(i),

where V(i) = V(rl(i) ) + V(q(i+l))

I 0 we need to compute the dual norm for

II ~(p~)II,

~ ~

L~(V)

relative to .Tp__.p. We need almost to repeat our computations for (4). Namely we need to apply the Parseval formula on (p, p) keeping in mind the Jacobian if(p) :

(19) II O(p,D ll2= c II ~ 11}4~ f

[

Im

sup

pEV* (rl)

~m

~(p,

7?d~) 9 ~(~; p) dp]

p=conat

For the same reason as for the inversion formulas the form which we integrate on 7 in (19) is closed. Therefore we can in the definition of the norm (19) integrate on any cycle homological to the regular cycle.

91 LECTURE 3. THE CONFORMAL RADON TRANSFORM In Lecture I we extended the affine Radon transform up to the projective R a d o n transform. It corresponds to the projective compactification of the affine space R " up to the projective space N P ' . In the process of the compactification we a d d to R" the line at infinity and extend the group of affine transformations up to the projective group SL(n + 1; R). There is another remarkable compactification of N '~ - the conformed compactification NC n. In this case we a d d only one point at infinity and extend the affine group up to the conformal group SO(n + 1; 1). T h e cop.formal space N C n can be realized as the sphera S n :

(1)

x~ + ' ' '

2 1 =1 -I" Xn+

The e m b e d d i n g of R n in NC n is the inverse m a p to the stereographie projection out of the north pole N = (0, . . . 0, 1) which corresponds to the point at infinity. Sections passing t h r o u g h N and only they correspond to hyperplanes. To see the action of the eonformal group we consider the projective coordinates in which S " corresponds to the quadric Q:

(1')

x n+l 2 =o

.....

and the group SO(n + 1; 1) of projective transformations conserving [](x) is exactly the conformal group. W h e n we transfer from the affine geometry to the conformal one we e x t e n d the family of hyperplanes up to the family containing all h y p e r p l a n e s as well all as spheras. This geometry contains as a subgeometry not only the affine geometry but also the hyperbolic geometry. Let us consider the affinization x,~+l = 1 where we o b t a i n the affine h y p e r b o l o i d Hn:

(2)

x2

_

x 2n = 1

x2 .....

and the hyperbolic geometry is realized on one of its two sheets H_~ (x0 > O) (or we can identify the symmetric points) with the group of hyperbolic motions SO(n; 1). T h e projection out of ( 0 , . . . , 0 ) to the plane x 0 = 1 gives the reaIization of the hyperbolic space inside the sphera. These geometrical facts will present us with the possibility to include the affine and the hyperbolic R a d o n transforms in a unified construction. 1. B a s i c d e f i n i t i o n s . Let f ( x ) 9 O ( - n + 2). Then we define the conformal R a d o n t r a n s f o r m as the integrals of f on the sections of Q by the h y p e r p l a n e s L~ = {(~, x) = 0):

(3)

?(,)

=s

=

= (t0,''',fn+i)

s f(x),() 9 R n + 2 \ { 0}

Here we need to know f in a neighborhood of Q for the first version of the definition and only on Q for the second one. Of course ] is nothing more t h a n the projective

92 R a d o n transform of the distribution f(x)5([~(x)) but we will see t h a t it deserves the special consideration. It affine coordinates we have

(3')

/ ( ~ ) = --~

. I(X)~((~, X))CO(~,I,...,Xn+I )

Correspondingly

f({) = ~

(3")

~(x0,...,,,)=

,, f(x)5(({, X))We(XO,...,Xn),

Z (-1)J-'~,xJA 0 (a point outside of the ball) then the essential part of Fy0 lies outside 7~ (some hyperplanes passing y0 do not intersect Q). The form ~J? will be for general f singular on these cycle. Nevertheless for a special class of functions we can to contract the integration on them to the integration on cycles in ~. Let ~/0 will be the polar for y0 relatively Q. It means that the points of the section of Q by the plane L(~ ~ = {(7/~ x} = 0} are the tangent points of the

97 tangent hyperplanes to Q through y0. Let us L(q ~ divides Q into 2 open parts Q• Let (17)

s u p p f C Q+

Then we can extend a / o u t s i d e E such a way that the cycle Fy0 will be homotopic in this extended domain to cycles in E and

(18)

~r g'][x] = 2c f ( x ) xO

We used (17) to reduce g to g'. For the the regulaxization we can use any plane passing through y0. We can to write the inversion formula if instead the condition (17) we suppose that f is zero in a neighborhood of L(q~ For the hyperbolic geometry we take y0 = (0, . . . , 0, 1). In affine coordinates (xn+l = 1) it will the point (0) ~md the plane L(r] ~ will be the hyperplane at infinity. We consider finite functions f on H~_ and the cycle Fx0 of hyperplanes through (0). This family is SO(n; 1)-invaxiant and sections axe hyperbolic hyperplanes. So (18) gives the inversion formula for the hyperbolic geodesic Radon transform. As the affinization ~ for the regulaxization usually it is taken t be the hyperplane through (0) invariant relative to the isotropy subgroup of the point x. We could see in the consideration of this example that for inversion formulas it is important not only to construct the operator n out of the image into closed forms but also to investigate carefully appropriate cycles. In this example there was a big invaxiance group but it was not really crucial. The construction of the operator ,r can be generalized in the case when instead of the quadratic form [3(x) we consider either another quadratic form or any homogeneous polynomial of any degree [G6]. However the choice of cycles is much more complicate in the more general situation and moreover appropriate cycles often do not existed. REFERENCES [FG]

[GG]

[GGG] laGS] [GGV] [G1]

[a2] [a3]

Faraut J., Gindikin S., Psedo-Hermitian Symmetric spaces of tube type, Topics in Geometry: In Memory of Joseph D'Atri (Gindikin S., ed.), Birkhauser, 1996, pp. 123-154. Gelfand I, Gindikin S., Nonloeal inversion formulas in real integral geometry, Funct.Anal. Prilozh. 11 (1977), no. 3, 12-19 (Russian); Engl.transl.: Funct.Anal.Appl. 11 (1977), 173-179. Gelfand I., Gindikin S., Graev M., Integral geometry in aj~ne and projective spaces,, Itogi nauki i techniki 16 (1980), VINITI , pages 55-224 (Russian); Engl.transl.: J.Sov. Math. 18 (1980), 39-67. Gelfand I., Graev M., Shapiro Z., Integral geometry on k-dimensional planes,, Funct.Anal. Prilozh. 1 (1967), no. 1, 15-31 (Russian); Engl.transl.: Funet.Anal.Appl. I (1967), 14-27. Gelfand I., Graev M., Vilenkin N., Integral geometry and representation theory,, Generalized Functions, vol. 5, Fizmatgiz, 1962 (Russian); Engl.transl.: Academic Press, 1966. Gindikin S., The Radon transform from cohomological point of view, 75 years of Radon transform (S.Gindikin, P.Michor, eds.), International Press, 1994, pp. 123-128. Gindikin S., Holomorphic language for O-cohomology and representations of real semisimpie Lie groups, Contemporary Math. 154 (1993), 103-115. Gindikin S., Integral formulas and integral geometry for "O-cohomology in CP n, Funct. Anal. Prilozh. 18 (1984), no. 2, 26-33 (Russian); Engl.transl.: Funct.Anal.Appl. 18 (1984).

98 [G41

[G5] [G6] [GH]

[GRS] [Gon]

Gindikin S., Unitary representations of groups of authomorphisms of Riemann symmetric spaces of null curvature, Funct.Anal. Prilozh. 1 (1967), 26-33 (Russian); Engl.transl.: Funct.Anal.Appl. 1 (1967), 28-32. Gindikin S., Fourier transform and Hardy spaces of-O-cohomology in tube domains, C.R. Aca~l.Sci.Paris 415 s~rie I (1992), 1139-1143. Gindikin S., Integral geometry on real quadrics, Amer.Math. Soc.Transl.(2) 169 (1995), 23-31. Gindikin S, Khenkin G., The Cauchy -Fantappie formula on projective space, Amer.Math. Soe.Transl.(2) 146 (1990), 23-32. Gindikin S., Reeds J., Shepp L., Spherical tomography and spherical integral geometry, Lectures in Applied Math. 30 (1994), Amer.Math.Soc., 83-92. Goncharov A., Integral geometry and W-modules, Math. Research Letters 2 (1995), 415435.

DEPARTMENT OF MATHEMATICS, HILL CENTER, RUTGERS UNIVERSITY, NEW BRUNSWICK, NJ 08903, U.S.A.

E-mail address: [email protected]

CIME Lectures Venice June 1996

RADON

TRANSFORMS

AND

WAVE EQUATIONS.

SIGURDUR HELGASON

1. I n t r o d u c t i o n . These lectures give a short account of the basic Radon transform theory on Euclidean and symmetric spaces. The applications are focussed on the wave equation with proofs. These applications are a mixture of old and new results, as described specifically in the text. Background material can, to a large extent, be found in my books [H9], [Hll] where some of the basic tools axe discussed in greater detail. I would like to thank the organizers, E.C. Tarabusi, M. Picardello and G. Zampieri, for inviting me to present these lectures. The following standard notation will be used. In a metric space X with metric d, C(X) denotes the space of continuous complex-valued functions, Br(x) denotes the open ball {y E X : d(x, y) < r} in X. For a manifold X, we use Schwartz' notation C(X) (resp. :D(X)) and T)'(X) (resp. C'(X)) for the space of C ~176 functions (resp. C ~176 functions of compact support) and distributions (resp. distributions of compact support). If X = R " , S ( R n) denotes the space of f c C(R '~) for which each derivative goes to 0 at oc faster than any negative power of the distance Ix[. For Lie groups the Lie algebra is denoted with the corresponding German letter, say G and 9. The adjoint representation of G is denoted Ad and of 9 by ad. 2. T h e F l a t Case. W a v e E q u a t i o n s . Let p n denote the set of hyperplanes in Euclidean space R% The Radon transform on R n assigns to each function f on R n, integrable over each hyperplane, the function f" on p n given by

(2.1)

:: ~ f(x)dra(x),

~ E pn,

where dm denotes the Lebesgue measure on ~. Along with the transform f ~ f" we consider the dual transform ~ ~ ~ which to a function ~ on P'~ assigns the function ~ on R n given by (2.2)

~(x) =: /

~(~)d#(~),

x E R n,

100

where d# denotes the average on the (compact) set of hyperplanes passing through X.

Representing a hyperplane ~ by the equation (x, w) = p where ( , 1 denotes the inner product, w a unit vector, and p E R, we see that P~ has a natural manifold structure. Since the pairs (w,p) and ( - w , - p ) represent the same ~, the mapping (w,p) ) ~ of S n-1 • R onto P~ is a double covering. Note that (2.2) can also be written

1/

~o(x) = -~

(2.3)

(p(w, (x,w) )d.w,

S,~-x

where dw is the volume element on S ~-1 and ~ its area. The inversion of the transform (2.1) can be expressed in various ways; for us the following formulation will be convenient. In analogy with the space $ ( R n) we consider the space S ( P n) of smooth functions on P " for which each derivative is rapidly decreasing in the p variable. We define the operator A by d~_i d--~=-r~(W,p),

(2.4)

do-,

7-lpd~=T~(w,p),

n odd n even

where 7-/denotes the Hilbert transform

(~F)(t) = ~i JR tF(P) - p dp, F e S ( R ) . Then (2.1) is inverted by (cf. [H2], ILl)

cf = (A f ) v

(2.5)

f e S ( R n)

where the constant c equals c = (-47r) ~

r(") r( 89

v

The transform T ~ ~ can be inverted similarly ([H4], [Hll], [Go], [So]) but this will not be needed here. As is well known, the Radon transform is particularly well suited to a study of the wave equation 02U

(2.6)

=

= f0(x),

=

where L = f i 0~ (0~ = 0/0x~) and fo, fl E 73(R"). Our treatment resembles t h a t i=l

of Lax and Phillips [LP1], CoL 2.1, p. 105 for n odd (with some simplifications) but applies to b o t h cases n odd and n even.

101 Lemma

If h C C2(lq.) and w C S n-l, then the function

2.1.

v(x, t) = h((x,w) + t) satisfies Lv = (Ot)2v. The proof is obvious. For a function ~ on S n-1 • R we now put 0~

0~ = ~ ,

Theorem

(2.7)

(n~)(~,p) = np(~(~,;)).

The equation (2.6) has the solution

2.2.

(0 ~ - ' ^fo + c ~ - 2 ~ ) (w, (w, x +) t) (dw n

~ ( z , t ) = cl

odd)

s,~-,

u(x,t)=c2

/

(7"t(On-lffo+On-2~))(w,(w,x}+t)dw

(neven)

s,~- i The constants cl, c2 are given by Cl = 89(21r) l - n , c2 = 89(27ri) 1-n

PROOF. (i) n odd. Because of Lemma 2.1 the right hand side does satisfy the wave equation so we just have to verify the initial data. For this note that the function ~

(0n-~s

(w, (w, x})is odd so u(x,0) = f0(x) follows from (2.5). Applying 0t

raises the derivatives in (2.7) by one and now the function w ~ (Onfo)(w, (w, x)) is odd so again ut(x, O) = fl(x) by virtue of (2.5). (ii) n even. Here we just have to use the commutation 07-t = ?-/0; otherwise the proof is the same as for the case n odd. It is also easy to write down the solution to (2.6) using the Fourier transform

Y(~) = f f(x)e-i(~'r . 2

It"

In fact, assuming the function x ~

(2.s)

u(x, t) in S ( R n) for a given t we get

~,,(r t) + (~, ~)~(~, t) = 0

and solving this ordinary differential equation we obtain (2.9)

~(~, t) = f0(~) cos ]~]t + j~ (~)sin [~]t

102

The function ( ) (sin [(It)/l(I is entire of exponential type Itl on C '~ (of at most polynomial growth on R n) so by the Paley-Wiener theorem on R ~ there exists a Tt E E ' ( R '~) such that sin [~lt - f/ e-i(r

(2.10)

.

R~

The following result is suggested by (2.9) but is readily verified even if fo and fl do not have compact support. Theorem

2.3.

Given fo, fl E $ ( R n) the function

(2.11)

u(x, t) = fo * T[ + fl * Tt

satisfies equation (2.6). Here T[ stands for Ot(Tt) and * denotes convolution.

C o r o l l a r y 2.4. If fo, fl have support in BR(O) then u has support in the region

]xl < Itl + R. In fact, the support of Tt is contained in Bltl(0 ) - so formula (2.11) shows that the function x , u(x, t) has support in (BR+Itl (0))-. Combining this with Theorem 2.2 we can deduce another classical result. C o r o l l a r y 2.5. (Huygens' principle). Let n > 1 be odd. A s s u m e fo and f l have support in BR(O). Then u has support in the shell (2.12)

Itl - R < Ixl ___ Itl + R

which is the union (2.13)

[_J

C~

IyI n.

If z E R '~, F C E ( R '~) we denote by F z the translated function y ~ F ( y + z). Then u z satisfies (2.6) with initial data f~),f~ which have support contained in BR+IzI(0). Hence by (2.14) uZ(O,t) = 0 for It] > R + ]zl, i.e. (2.15)

u(z, t) = 0 for Itl > n + Izl.

In view of Cor. 2.4 this proves (2.12).

103 Finally, if ( x , t ) 9 Cy with lYl -< R we have I x - y l : [tl so Ix[ _< Ix-yI+lyl A

f(x)=O

for

d(o,x) > A.

we have

The proof uses the fact that the geodesic spheres for the metric (4.1) are ordinary spheres and secondly the metric (4.1) is conformal to the flat metric on R n. This second property was used by Kurusa [K] to relate the transform f ~ f" to the flat

107

k-plane transform on R n. Such a relationship can be used to describe the range for the transform f ----* f" (see [BT], [I] and for k = n - 1 [Gi]). C o r o l l a r y 4.3.

The X-ray transform on X is injective on the space of functions

satis/ying (4.6). This holds also for the symmetric spaces X = G / K considered in the next section ([H8], p. 133). 5. R a d o n T r a n s f o r m o n a S y m m e t r i c S p a c e . Let X = G / K be a symmetric space of the noncompact type, i.e. G is a connected noncompact semisimple Lie group with finite center and K a maximal compact subgroup. Let g and ~, denote their respective Lie algebras. Let o denote the origin e K in X. We consider the Iwasawa decompositions G = N A K , g = n + a + ~ where N and A are nilpotent and abelian respectively. We write accordingly g = n(g)expA(g)k(g),

A(g) e a.

A horocycle ~ in X is an orbit Ng 9 x of some point x E X under a conjugate Ng = gNg -1 of N. Let -E denote the set of all horocycles ~. The group G acts transitively on ~ and the subgroup which maps the horocycle ~0 -- N . o into itself equals M N , where M is the centralizer of A in K. Thus (5.1)

-E = G / M N = ( K / M ) • A.

The Radon transform on X maps functions f on X into functions f on -E by (5.2)

f'(~) -- j~ f ( x ) d m ( x ) ,

~e

where dm is the Riemannian measure on ~ up to a constant factor (independent of ~). In group-theoretic terms, (5.3)

f'(g" ~0) = / f ( g n . o)dn

(g 9 G)

N

and here the Haar measure dn on N is normalized by the relation e2p(A(~

= 1

N

where 9 is the Cartan involution of G with fixed point set K and (5.4)

2p(H) = T r ( a d g l n ) ,

g 9 a.

Since ~ = ( K / M ) • A we sometimes write r

~o) = ~(kM, a)

108

if ~ is a function on ~. The dual transform ~ ~ (5.5)

~(x) = f

V .

~ is given by

~(~)dp(~)

the average over the space of horocycles passing through x. terms,

(5.6)

. o) = [

In group-theoretic

(gk . o)dk.

K

To put this in a form analogous to (2.3) we introduce a vector-valued analog of (x, w} in (2.3). For x = gg 9 G/K and b = kM in K/M we put

A(x, b) = A(k-lg) 9 a.

(5.7) Then ([H5])

(5.8)

~(x) = /

~(kexpA(x, kM).~o)e2p(A(x'kM))dkM

K/M

where

K/M

dkM is the normalized K-invariant measure on K/M. We also write B for db for dkM. For A 9 a*, the complex dual of a, we consider the spherical

and function

(fix(g) = /

e (ix+p)(A(kg))dk.

K

If a + C a is the positive Weyl chamber and a*+ the corresponding chamber in a* it follows from [HC], p. 291 (see [Hb], Ch. IV, w that the limit (5.9)

c(A) =

lim e(-ix+p)(tH)9)j~(ex p t H )

t~+co

exists provided Re(i;~) 9 a*+ and the limit is independent of H 9 a +. From the formula for this c-function ([GK]) in terms of G a m m a functions it follows that it extends to a meromorphic function on a* and it is easy to prove that 1/c(A) has each of its derivatives bounded by a polynomial on a* ([H3]). Consider now the Euclidean Fourier transform on A (5.10)

F*(),) =

f F(a)e- x( ~

.X 9 a*.

The measures da and dA are normalized such that the inversion formulas hold without a multiplicative constant. We then define the operators j and j on ,S(A) by

(iF)* (:,) = c(A)-I F* (:9 (jF)* (:~) = c ( - ~ ) - 1F* (~)

109

and the operators J and J by (J~)(kaMN) = ja(~(kaMN)) ( ] ~ ) ( k a M g ) = ~a(~(kaMN))

the subscript a denoting the variable on which j acts. Finally we put A = e - P J o eP, A = e - p ] o eP where eP(kM, a) = ep0~ The transform f ~ f" is inverted by the following result. ([H2], [H5], [Hll]).

Theorem 5.1.

With A and A defined as above

(5.11)

f = w-l(hAf) v

f 9 7)(Z)

where w is the order of the Weyl group W of X .

For G complex another inversion is given in ([GG]). The operators A and .~ are differential operators precisely when G has all its Cartan subgroups conjugate. R a n g e s a n d kernels. The Radon transform f ---* f can be extended to all f 9 L I ( X ) and then it is still injective [H5]. The range :D(X) ^ can also be described explicitly ([H6]) and it implies the surjectivity, E(E) v = E(X) for the v dual transform ~ ~ ~. However this last transform has a huge kernel so there is no useful inversion formula for it (in contrast to R n) except under the condition of K-invarianee ([Hll], V, w E x a m p l e . To illustrate the description of the range mentioned for the map f ~ 7 and the kernel of ~ ~ ~v we consider the simplest case X = H2 (without the factor 4 in (4.1). The horocycles are the circles internally tangent to Ix I = 1. Let ~t,e denote the horocycle through eie, with non-Euclidean distance t (with sign) from 0. A function r 9 E(~) is determined by its Fourier series

(5.12)

r

= E

Cn(t)e~n~

r

9 E(R).

nEZ

Theorem 5.2. (i) The range D(H2) A consists of the functions (5.12) such that for each n C Z, (5.13)

Ca(t) = e-t ( ~ t - 1 ) ( d _ where ~

3)... (d-

2In] + 1 ) ~ n ( t )

E Z)(R) is even. v

(ii) The kernel of ~ --~ ~ on E(E) consists of the functions (5.12) such that for each n C Z,

(5.14)

(d+l)

(d+3)...(d+2,n,-1)etr

isodd.

110

R e m a r k . For X = R 2 the analogous result is the following: Let ~t,o denote the line (x, e i~ = t and expand a function r on p2 by (5.12). Then r E :D(R 2) if and only if for each n, (5.15)

dlnl = d--~Tn(t),

r

T~ E D ( R ) even

and r E E ( P 2) is in the kernel of the dual transform if and only if for each n,

dtnl dtin ICn(t)

(5.16)

is odd.

As a part of the range characterization of D ( X ) ^ one has the following support theorem ([H6]). T h e o r e m 5.3. Let B C X be a closed ball. If f E 7P(X) and ff(() = O for MB=O then f(x)=O forx• B. This still holds with f sufficiently rapidly decreasing. The proof in [H6] is based on studying K-types; a different proof was given by [GQ]. 6.

The Fourier Transform on X = G/K.

T h e Fourier transform on R n can be written

f(Aw) = / f(x)e-i'~(x'~)dx. R=

To define a Fourier transform on X = G/K we replace (x, w) by the vector-valued inner product A(x, b) in (5.7). Thus we put for a function f on X,

f(A, b) = f f(x)e(-i;~+P)(A(x'b))dx

(6.1)

X

for all A E a~, b E B for which the integral converges. Here dx = dgK is the volume element on X, defined by the Haar measures dg and dk. The measure dg is normalized by

f f(g)d9= f G

f(nak) e-2p('~

NAK

in the notation of w Theorem (6.2)

6.1.

The Fourier transform is inverted by

1 / f(x) -----w

f(A,b)e (i;~+p)(A(x'b)) dAdb

ic(A)] 2 ,

f E D(X).

a*xB

Moreover, f ~

f extends to an isometry of L2(X) onto L2(a~_x B; ]c()~)]-2dA db).

111

If f is K - i n v a r i a n t this reduces to H a r i s h - C h a n d r a ' s m a i n results for the spherical transform on X which is the s y m m e t r i c space analog of t h e Bessel t r a n s f o r m (X)

F(%) = j F(r)Jo(rA)dr. 0

For applications to differential equations it is useful to have a P a l e y - W i e n e r t y p e theorem, t h a t is an intrinsic characterization of the space T)(X)':. Here the result is as follows ([H6]). T h e o r e m 6.2. The range T)(X)~ is given by the space of/unctions ~ E $(a* • B) satisfying a) /~ ~ ~(%, b) is an entire function of exponential type, uniformly in b. b) The integral

/

~(,1, b)e(i~+P)(A(~'b))db is W-invariant in )~.

B

7. T h e W a v e E q u a t i o n o n X = G / K . Denoting the Laplace-Beltrami o p e r a t o r on X by L x we consider t h e shifted wave equation

(Lx + Ipl 2) ~ -- ~Ot2

(7.1) with Cauchy d a t a (7.2)

u(x,O) : fo(x), ut(x,O) = fl(x),

f o , f l E l)(X).

T h e shift by Ipl 2 is related to t h e curvature t e r m in (3.1). In this section we assume r a n k X : 1. Let H0 be the unit vector in t h e positive Wely chamber a +. Since the c-function c(,~) is singular at ,~ = 0, the o p e r a t o r s J and J can b e factored by

J = HoJ',

J = H0J',

where J ' and J ' are new operators on functions on E. Theorem

The solution to the Cauchy problem (7.1)-(7.2) is given by

7.1.

1

^ B

Remark.

For X -- H 3 this agrees with formula 3.41 in [LP2].

112

We prove this by means of a couple of lemmas.

L e m m a 7.2. Let F E C2(A), b C B. Then the function v(x, t) = F(exp(A(x, b) + tHo))e -Iplt

(7.4)

satisfies (7.1). PROOF. Because of the K-invariance of the equation (7.1) we may assume b = eM. Then both (7.4) and (7.1) are N-invariant so it suffices to verify (7.1) on A . o . The restriction of L x + [p[2 to N-invariant functions is on A 9 o given by the N-radiM part A N ( L x + Ipl 2) which equals ePLA o e-P (see [H9], Ch. II) where LA is the Laplacian on A- o. In our case, LA = H3 so the problem is to verify

)

(ePH2 o e-P)(F(aexp(tHo)))e -Iplt = Or-~ F(aexp(tHo))e -Iplt . Using p(Ho) = IPl this is a routine matter. L e m m a 7.3.

(7.5)

For f E •(X)

we have the identity

/ (J1])(eP f)(b, exp A(x, b))ep(A(='b))db = O. B

The same identity holds with j i j replaced by HoJ]. PROOF. First recall the relationship between the Radon- and Fourier transforms, valid for X of any rank, (7.6)

f(A, b) ---- f(ePf')(b, a ) e -i)~(l~

a)da

A SO

y(A, b) 2

= f J'2'(ePf)(b' a)e-i~O~ A

Using the invariance condition in Theorem 6.2, b) for x = 0 we deduce that

(J'J') (ePf) (b, a)db B

is even in a and consequently,

/ J'](ePf~(b, e)db = 0

B

113

being the derivative at a = e of an even function on A. This proves the lemma for x = o. On the other hand, A = e - p J o e p,/~ = e-P J o e p and if we define A ~ -- e - P J ~ o eP the expression in the lemma is ( A ' A f ' ) v (x). However, the operators ^,v,~., A' are all G-invariant ([H5], p 42) so since the expression (7.5) vanishes for x = o it vanishes identically. This proves the lemma since the statement for H o J J follows in the same way. PROOF OF THEOREM 7.1.

In the integrand in (7.3) we replace e p(A(z,b)) by eP( A(x,b)+tHo) e-lPlt"

Then for b fixed the integrand has the form (7.4) so the right hand side of (7.3) certainly satisfies the wave equation (7.1). It remains to check the initial conditions. We put t = 0 in the right hand side of (7.3). Then the term containing fl vanishes by virtue of L e m m a 7.3. Introducing A = e - P J o e p, A = e - p ] o e p and making use of (5.8) we see from the inversion formula (Theorem 5.1) that the right hand side of (7.3) reduces to f o ( x ) . Hence u ( x , O) = f o ( x ) . For the initial condition u t ( x , O) = f l ( x ) we apply (OlOt)o to (7.3), apply L e m m a 7.3 for H o J J and deduce that the term in (7.3) containing f0 gives no contribution to u t ( x , O ) . Again we deduce u t ( x , O) = f l ( x ) from the inversion formula. This proves Theorem 7.1. C o r o l l a r y 7.4.

The solution to the Cauchy problem ( 7 . 1 ) - ( 7 . 2 ) can be w r i t t e n

(7.7) K

In fact, if g = e this agrees with formula (7.3) (with x = o). By the G-invariance of (7.1) a translate by g C G in the functions f0, fl implies a similar translate in the solution u. Since the operators f ---o f', A,/~, A ~ are all G-invariant formula (7.7) holds in general. Remark. Theorem 2.2 could also be stated in the form of (7.7) but then we would have to consider the orientation preserving isometry group of R n as acting on S '~-1 • R , not on p n . This difference between R n and G / K of rank 1 can be traced to the fact that our hyperplanes in R n are not given an orientation whereas for G / K we always have two horocycles passing through a given point with the same tangent plane. Other quite different explicit solution formulas for the wave equation (mostly in terms of spherical averages) can be found in [P], [BO], JIM] for X of rank 1 (using the classification) and in [CY] and [Hll]V, w for more general X. In fact the solution to (7.1), (7.2) for X of any rank can be written ([Hll], V, w

(7.8)

u ( x , 0 = (/0 x E;)(x) + (/1 x &)(x)

114

where gt E g ' ( X ) satisfies (7.9)

(Lx + Ipl ~) s~ = ~-~t~s~

(7.10)

go = 0, E~ = 60

g~ denoting a g t. Moreover g.t has Fourier transform given by ~(A, b) - sin(IA[t)

and thus has support in the ball Bltl(O)-. In (7.8) x denotes the convolution on X induced by the convolution product on G. H u y g e n s ' P r i n c i p l e . The operators J, J are differential operators exactly when c(A) -1 is a polynomial. For X of rank 1 this happens only for the odd-dimensional hyperbolic spaces H ~. C o r o l l a r y 7.5. Let X = H 2n+l. Suppose fo and fl in (7.2) have support in the ball B~(O). Then the solution u to (7.1)-(7.2) is supported in the conical shell

Itl

- R R + d(o,g, o) whence (7.12)

u(y,t)=O

if

]tl>_R+d(o,y ).

On the other hand (7.8) shows that x ---* u(x,t) has support in BR+ltl(O). This and (7.12) proves the corollary. 8. T h e M u l t i t e m p o r a l W a v e e q u a t i o n o n S y m m e t r i c Spaces. We consider again the symmetric space X = G / K and the algebra D ( G / K ) of G-invariant differential operators on X. From [HC] w we consider the isomorphism F : D ( G / K ) ---* I(a) where I(a) is the set of W-invariants in the symmetric algebra S(a). Given p E S(a) we let O(p) denote the corresponding constant coefficient differential operator on a. Each c0(p) operates on the symmetric algebra S(a*). Let H(a*) C S(a*) be the subspace (the space of harmonic polynomials) annihilated by all a(p), p E I(a), without constant term. Then dimH(a*) = w ([HC], w

115

Let pl = 1 , p 2 , . . . ,pw be a real homogeneous basis of H(a). We consider now the following system for a function u 9 E(X • a): (8.1)

Du = 0 ( r ( D ) ) ~

for an D e D ( G / K )

with initial data

(8.2)

(O(pr

=fi(x),

1 on it with values in

a"b"

(a, b) = Z

a, b e C(S(a))

a6W

where a~(;Q = a(a-1)~) ()~ 9 a*). Define qJ by (8.3)

(qJ,Pi) = 5ij,

1 O. Then all CR functions on M extend locally to the side r(z) < 0 near p. Tr~preau [Tr] found a necessary and sufficient condition for the one-sided extendibility. Theorem

1.2. (Tr6preau [Tr]) Let M be a hypersurface in C N and let p 9 M.

Assume there is no complex hypersurface S C M passing through p. Then all CR functions on M holomorphicaUy extend to the same one-sided neighborhood of p. Conversely, if such a hypersurface S exists, then for every neighborhood U C M of p there is a CR function f in U that does not extend to any one-sided neighborhood ofp. We will o b t a i n these theorems as special cases of a more general result. We now consider manifolds of higher codimension.

Let M be a s m o o t h real

submanifold in C N. We define the complex tangent space at p 9 M the same way as for hypersurfaces:

Tp(M) -- Tp(M) M JTp(M), p 9 M. T h e manifold M is called a CR manifold if dim T~(M) does not d e p e n d on p E M . T h e manifold M is called generic if Tp(M) spans Tp(C N) -

C N over C for all

p E M , t h a t is

Tp(M) + JTp(M) = C g. For example, all real hypersurfaces are generic. If M is generic, then M is a C R manifold a n d d i m e Tp(M) + codim M = N, where c o d i m M is t h e codimension of M in C N. T h e dimension d i m c T ~ ( M ) is called t h e C R dimension of M and is denoted by C R d i m M . W e define C R functions on M the same way as for hypersurfaces: a C 1 function f is a C R function if df is C-linear on Tp(M) for all p E M or equivalently, df A

dZl A . . . A dzg [M of distributions.

= 0. We a p p l y this definition to continuous functions in t h e sense

126 2. T h e B a o u e n d i - T r e v e s

approximation

theorem

T h e celebrated Baouendi-Treves theorem states t h a t a C R function locally can be a p p r o x i m a t e d by holomorphic polynomials.

Theorem

2.1. (Baouendi and Treves [BT]) Let M be a generic manifold in C N.

Then for every point p E M there is a neighborhood U C M of p such that for every continuous CR function f on M there is a sequence of polynomials f~ such that f;~lv converge uniformly to f l u as ~ --* oo. Proof. Let Mo be a m a x i m a l l y real submanifold t h r o u g h p, t h a t is p E Mo, TC(Mo) = 0, a n d d i m Mo = n. We introduce coordinates in C y in such a way t h a t p = 0 a n d

Tp(Mo) -= R N C C N. Shrinking M0 if necessary, we assume t h a t there exists 0 < a < 1 so t h a t for all distinct z, w E M0, we have

(2.1)

IIm (z - w)[ < ~ IRe (z - w)l.

We introduce t h e entire functions

f~(z) = ()~/Tr)g/2 f

f ( w ) e - ~ ( z - ~ ) 2 d w i A . . . A dWN,

JM o

where (z - w) 2 : = ~ ( z j - wy) 2, ,~ > 0. The condition (2.1) ensures t h a t

(.k/z~)g/2e-)~(z-w)2dWl A ' " A dWN form a 5-shaped sequence as ,k --* exp. Thus

f~(z)~f(z)

for

z9

as

,~-~co.

We prove t h a t f~ --~ f in a neighborhood of p 9 M. Let us view M0 as a manifold with boundary, a n d let M1 be a slight p e r t u r b a t i o n of Mo with the same boundary. ~Ve define ]~ = f M : ' ' "

by integrating the same expression as in f~. T h e n by

the same argument, ];~(z) ---* f ( z ) for z 9 M1. But actually ]~(z) = f~(z) for all

z E C g . Indeed, M0 and M i b o u n d a submanifold Mol C M , OMoi = Mo - Mi. Since e -~(z-~)2 is holomorphic and df A dwi A ... A d w g l M -~ 0, the integrand is a closed form on M . By the Stokes formula f~ - s

= fMo -- fM: = fMol = 0. Thus

f~ converge to f on every p e r t u r b a t i o n M i of Mo, hence in a neighborhood of p on M . To a p p r o x i m a t e f by polynomials, one takes the Taylor polynomials of f~. T h e t h e o r e m is proved.

127

3. Bishop's equation Let A C C b e t h e s t a n d a r d unit disc A _-- {~ C C : I~1 < 1}, and let bA = {~ C C : I~1 = 1} be the unit circle. A n analytic disc in C g is a m a p A : A --* C N holomorphic in A a n d s m o o t h up to the boundary. Let M b e a manifold in C N. We say t h a t the disc A is a t t a c h e d to M if A m a p s the circle bA to M . Let f b e a C R function on M and let A be a sufficiently small analytic disc a t t a c h e d to M . T h e n f is a limit of a sequence of polynomials f~ t h a t converges to f on bA := A ( b A ) c M . Since the functions f ~ o A converge on bA, by the m a x i m u m principle, t h e y converge inside A. Thus f~ converge on the set A ( ~ ) . a set filled up by small discs a t t a c h e d to M .

Let ~ be

T h e n the sequence of holomorphic

functions f), converge uniformly on ~t. If ~ is an open set in C N, then the limit of f~ is holomorphic. If ~t is a C R manifold, then the limit is a C R function. In 1965 E. Bishop [Bi] introduced a construction of analytic discs a t t a c h e d to a real manifold M c C g.

Proposition 3.1. Let M be a CR manifold in C~N = C n • C "~ and let p E M . Assume that the projection w : C g --~ C n maps T ~ ( M ) isomorphicalIy to C n. Then for every q C M close to p and every sufficiently small analytic disc r : A ---* C n with r

= w(q) there exists a unique analytic disc A attached to M such that

woA=r

andA(1)=q.

Proof. Let p -- 0.

We choose w to be p a r t of coordinate functions in C N a n d

complete it to a system of holomorphic coordinates (z = x + iy, w) E C " • C n. We first assume M is generic. T h e n we can choose the z coordinates so t h a t T~(M) has t h e equation x = 0, so T p ( M ) has the equation z = 0. T h e n M has a

local equation x = h(y, w),

where h is a s m o o t h function with h(0) = 0 and dh(O) = O. Let ~ ~-* A(~) = (x(~) + iy(~), w(~)) be an analytic disc in C g . T h e disc A is a t t a c h e d to M if a n d only if A(~) E M for ~ E bA. T h a t is (3.1)

x(0 = h(Y(0, w(0),

M = 1.

Since x(~) a n d y(~) are harmonic conjugates, they are related by t h e Hilbert t r a n s f o r m on bA, t h a t is y = T x + c, where T is the Hilbert transform on t h e unit

128

circle bA, and c = ~-~

)dO = const. By (3.1) we have

(3.2)

y --- T h ( y , w) + c

The equation (3.2) is Bishop's equation. Given the analytic disc ~ ~-~ w(~) in C n and c E R TM, the solution ff ~-* y(ff) defines the disc A by harmonically extending z(~) = h(y(~), w(~)) + iy(~) from bA to A. Since our disc is specified by the condition A(1) -- q, we modify the equation in the following way. For a function ~b on bA, we set T1r := T r - (Tr

The

function T1r is characterized by the conditions 1)

r + i T1r extends holomorphieally to A;

2)

(T1r

= 0.

Then (3.2) turns into y = T l h ( y , w) + y0,

where Yo = y(1). Given ~ ~-~ w(ff) = r

and Y0 = (the y-component of q) E R TM,

the solution defines the disc A. The existence and uniqueness of the solution to Bishop's equation follow by estimates in the Lipschitz space Ck,~(k _> 1, 0 < a < 1) because the term T l h ( y , w) is small. (See [Bo] and [T2].) We now consider the case in which M is not generic. Then we split the z coordinates z = (z', z") so that the projection (z, w) --~ (z', w) takes M to a generic manifold M ~ C C g' and the projection M ~ M ~ is a C R diffeomorphism, t h a t is it maps T O ( M ) isomorphically to TO(Mr). The inverse mapping M ' ~ M is given by the C R function z ' . By the first part of this proof, there is a unique disc A ~ attached to M I with w-component ~ ~ r

and A'(1) = (z'(q), w(q)). By the Baouendi-Treves approx-

imation theorem, the C R function z" extends to be holomorphic inside the disc A 1. The extension gives the z" component of the needed disc A. The proof is complete. For future reference we include the following. C o r o l l a r y 3.2. Let M be a real not necessarily CR manifold in C N = C n • C m, and p E M .

A s s u m e that f o r every q E M the projection w : C N --~ C n maps

T q ( M ) injectively to C ~. Then f o r every q E M close to p and every sufficiently small analytic disc r

A --. C n with r

= w(q) there is at most one analytic disc

A attached to M such that w o A = r and A(1) = q.

129

Proof. Note that dim T~(M) is an upper semicontinuous function of q, that is it can only j u m p down at a nearby point. Thus, without a loss of generality we assume t h a t dimc T~(M) = n. As in the previous proof, we choose w to be part of coordinate functions in C N and complete it to a system of holomorphic coordinates (z, w) E C m x C a. We again split the z coordinates z = (z', z") so that the projection (z, w) --* (z', w) diffeomorphicaUy maps M to a generic manifold M t C C N'. Then if there were two discs attached to M with the indicated properties, then their projectons to C N' would violate Propostion 3.1. The proof is complete.

4. T h e e d g e - o f - t h e - w e d g e t h e o r e m Let M be a generic manifold in C N. Let Np(M) := Tp(CN)/Tp(M) be the normal space to M in C N. The spaces Np(M) form the normal bundle N ( M ) . Let F be an open cone in Np(M). Let U be a neighborhood of p in M. We can identify F with a cone in a transversal plane H through p, H @ Tp(M) = C N. A

wedge W with direction cone F is a set of the form

w= ((Mnu)+r)ng. T h e following version of the edge-of-the-wedge theorem is due to Ayrapetian and Henkin [AH].

T h e o r e m 4.1. Let M C C N be generic, p C M. Let Mj 1 < j

<m=codimM

be manifolds with boundary M, in particular, dim Mj -- dim M + 1. Let ~j C Tp(Mj)/Tp(M) C Np(M) point inside Mj and let ~j form a basis in Np(M). Let F :-- C o n v { ~ l , . . . , ( m } be the convex span of ~,...~,~. Then all continuous CR m functions on M U Uj=I Mj extend holomorphically to the same wedge with direction

cone F', where F' C F is any finer cone, that is F ' \ { 0 } C F. m

Proof. We will show t h a t analytic discs attached to X = M U Uj=I Mj fill up a wedge W with edge M and direction cone F I. Then by the Baouendi-Treves approximation theorem, all CR functions on X extend to be holomorphic in W, what we need. Let p = 0. We introduce local holomorphic coordinates (z, w) E C m x C n = C N and define M and My (1 < j _< m) by a parametric equation

x = h(y, w, t),

130

where t C ~ m , and h is a s m o o t h Rm-valued functoin in a n e g h b o r h o o d of 0 E Rm • C n • Rm. T h e n the manifold M has the equation x = h(y, w, 0) while M j is defined as tj > 0, tk = 0 for k ~ j . We choose h so t h a t h(0, 0, 0) = 0, where

by,

hy(0, 0, 0) = 0,

h~(0, 0, 0) = 0,

ht(O,O,O)=I,

h~, and ht denote the derivatives of h, and I is the identity matrix. In

these coordinates the cone F turns into ] ~

:= {t E ]~m : tj _> 0, 1 _< j _< m}.

m Mj. Let ( H A ( ( ) = (x(() + iy((), w(()) be a disc a t t a c h e d to X = M U ~Jj=l

Note t h a t (y, w, t) form a set of local coordinates in a neighborhood of 0 E C g . Let ~-* t ( ( ) be t h e t-component of the discs A in the coordinates (y, w , t ) . T h e fact t h a t A is a t t a c h e d to X is expressed as follows. Let t ( ( ) = ( t l ( ( ) , . . . , tm(()). T h e n all ty(() _> 0 for ](] = 1, b u t for every ( E bA only one of tj-s can differ from 0. m We divide the circle bA into disjoint arcs bA = Uj=I-~j a n d want A to m a p ~,j

to My. Let Cy be a s m o o t h function on bA such t h a t Cj(() > 0, Cy(() -- 0 outside yj. We define for ( E bA t(~) = (/~l(~l(~),...,)~rn~)m(~)),

where A = ( - ~ 1 , . . . , / ~ m ) C ]~m. We take w ( ( ) = wo = const and y(1) = Yo C R m. T h e n the disc A satisfies the Bishop equation (4.1)

y = Th(y, Wo, t) + Y0-

T h e disc A depends on p a r a m e t e r s Yo, wo and A C 1~'~. We identify (Yo, Wo) with the point (h(yo,wo,0) + iyo,wo) E M. The disc A is a t t a c h e d to X if and only if

We consider the evaluation m a p

F : M xR'~---+CN ~ M x R ~ F : (Y0, w0, ~) ~ A(0) = (x(0) + iyo, w0), the center of A. T h e m a p p i n g F is defined in a neighborhood of 0 E M • R "~. T h e set W = F ( M x R TM) consists of the centers of discs a t t a c h e d to X. We will show t h a t the derivative F~(0) of F as a m a p p i n g M x R "~ ~ M • R m has m a t r i x

131

where the asterisk denotes unimportant elements. Hence, W contains a wedge with direction cone F' for every cone F ' finer than F - R + , what we need. __

m

Note t h a t for A = 0, the solution of (4.1) is y - 0, so A(() = (x0 + iyo,wo) and

x(r = h(yo, To, 0). Thus F[Mx{O} = id[Mx{O}. For general A we have A(0) = (x(0) + iy(O), To), x(O) = Pox, where 1 f2~ P o e = ~ Jo r176

(4.2) is the mean value of r

We now find Fx = OF/OA. We have

Fx(O, O, O) = (Pox~, 0), AtA=0,

w0=0,

Poxa = Po(hyyx + hits).

yo=0wehavehy=0,

tA =

0)

hi=I, "..

0

,

r

so PoX~ = Pot~. We choose the Cj-s such that PoCj = 1. Then F~(O,O,O) = (I, 0). Thus F~(O) has the needed form, which completes the proof.

5. L e v i f o r m a n d t h e w e d g e - e x t e n d i b i l i t y The H. Lewy extension theorem has a generalization to manifolds of higher codimension. In the final form, the result is due to Ayrapetian and Henkin [AH] and Boggess and Polking (see [Bo]). Let M C C N be a generic manifold in C g. Locally M is given by a local equation r ( z ) = 0,

where r = (rz,. 9 9 rm) is a smooth Rm-valued function in a neighborhood of 0 E C N such t h a t Or1 A... AOrm ~ O. Let H(p)(X, Y) be the complex Hessian of r, t h a t is

N j,k=l

n(p)(X,7) = ~

C~2T Xj~kk E C m,

OzjO~k

where X , Y e T~(M) C C N. We identify Np(M) with ]~m by dr(p): Np(M) --* R m, the differential of the defining function. Let V : = C o n v { g ( p ) ( X , X ) : X e Tp(M)}. The cone F C Np(M) is called the Levi cone of M at p. One can show that this definition does not depend on the choice of the defining function r.

132

T h e o r e m 5.1. Let M be a generic man]old in C N, and let p 9 M . Suppose the Levi cone F C N p ( M ) has nonempty interior. Then for every finer cone F ~ C F, that is F'\(O} C F, there exists a neighborhood U of p in C N such that all CR ]unctions on M extend to be holomorphic in the wedge W = ( (U M M ) - F') M U with edge M and direction cone - F ~. Proof. We introduce coordinates (z = x + iy, w) E C TM x C n = C N so t h a t p = 0 and M has local equation

r = x-

h(y,w) = O,

where h(0) -- 0 a n d dh(O) -- O. T h e n Tp(M) and T~(M) are defined by the equatons x -- 0 a n d z = 0 respectively. Let a E T~(M) ~- C n. We will construct a one parameter family of discs ~ ~-~ A((, t) attached to M so t h a t the centers of the discs trace a smooth curve t ~-* A(0, t), t > 0 in the transversal plane II = {(z, w) : y = 0, w -- 0} --- Np(M). We will show t h a t the curve has direction vector

d t=oA(O,t ) dt

-2H(p)(a,E)

at p. Repeating this construction for all nearby points q E M yields a smooth family of curves. Let Ma be a u n i o n of these curves. T h e n ]Via is a smooth manifold with part of its b o u n d a r y in M. Since Me contains the curve through p, then

TpMa has

direction -H(p)(a,-5) 9 Np(M). T h e n the theorem will follow by Theorem 4.1, the edge of the wedge theorem applied to all Ma. We now construct the needed family ~ ~ A(~, t), t > 0 by Bishop's equation y = Th(y, w).

(5.1)

Since we want A(0) 9 YI-- {w = 0, y = 0}, we take c = y(0) -- 0 i n (3.2). We further take ~ ~-~ w(~) in the form

w(r = act, where a E C n --- Tp(M). T h e n A(0) = (x(0), 0) e II, and x(O) -- Pox, where P0 is defined by (4.2). The centers A(0) -- A(0, t) form a smooth curve. We now find its direction at t -- 0. We denote by the dot the derivative with respect to t. We have

J:(O) -- Po(hyy + h,,(v + h~w). Note t h a t for w ----0 the solution of (5.1) is y - 0, so at t -- 0 we have w -= 0, y -- 0, x - 0. Therefore hy = 0, hw -- 0, and hence 4(0) = 0 at t = 0. Thus A(0,0) = 0

133

and the direction of the curve t ~-~ A(0, t) is determined by the second derivative A(0, 0). We have

~/ --- T(hu~/ + h~o(v + h ~ ) . At t = 0 the expression after T is identically equal to 0, hence Z) - 0 at t -- 0. We have

2(0) = Po(hy~) + hyyyy + 2h~ywy + 2h~ywy + hwwzb~b + h~--~--~ + 2h~w~hw). At t = 0 the partial derivatives of h are evaluated at (0, 0, 0), so they are constant. Sincehy = 0and9

= 0 a t t = 0, then all terms w i t h y o r

~) are zero. We have

zb = a~, so the term hww(v(v is holomorphic in r and vanishes at ~ -- 0. Then its mean value is Po(h~ow~b~b) -- 0. The same applies to the antiholomorphic term. Recall r = x - h(y, w). We have

~(0) It=o = 2hw~a-d Po ( ~ ) = 2hwwa-d -= - 2 H ( p ) ( a , ~), so the curve has direction -2H(p)(a,~) at t = 0, what we need.

The proof is

complete. We recall the intrinsic definition of the Levi form. Let C T ( M ) = T ( M ) | C be the complexified tangent bundle to M and let T1,0(CN) the bundle of (1,0) vectors in C g.

CTC(M) = TI,~

+ T~

Then TI,~

where T~

L ( p ) ( X , Y- -) : = ~1[ X , Y](p)

~---

{~-~.j=laj_~..~j}N 0 be N) n C T ( M ) , and

= TI'~ = TI,~ We set mod CTp(M),

where X , Y

are sections of TI,~ One can show that this definition depends only on X(p), Y(p). The Levi form n(p) is a bilinear hermitian form on TpX'~ with values in (TB(M)/Tp(M)) | C. Note that J : C N ---* C N the multiplication by i = x/-L~ identifies T p ( M ) / T p ( M ) with Np(M). The relation between n(p) and H(p) the intrinsic and extrinsic definitions of the Levi form is given by (see [Bo]) dr(p)JL(p) = H(p). In view of the intrinsic Levi form, the condition that the Levi cone F has n o n e m p t y interior F is equivalent to the condition that the commutators [X, Y] (p) of sections of TI'~

span Tp(M).

We say that M has finite type k at p E M if the tangent space Tp(M) is spanned by the commutators

[... [xlE1 ,x2s ] , . . . , x k ~k ](p),

(5.2) where Xj are sections of

TI'~

is the ease of finite type 2.

O

and X; j is either Xj or X~. The ease of r # 0

134

6. Minimality Let M be a C R manifold in C N. We say t h a t M is minimal at p E M if there is no p r o p e r C R submanifold S C M through p such t h a t C R d i m S = C R d i m M . Since S is only locally closed in M , the word "proper" means t h a t dim S < dim M . T h e m i n i m a l i t y condition was first introduced in IT1]. Note if M has finite t y p e at p C M, then M is minimal at p. Indeed, if S C M with the i n d i c a t e d properties exists, then X j in the definition of finite t y p e can be restricted to S, the c o m m u t a t o r s (5.2) are in CTp(S), so t h e y cannot span the whole space Tp(M). In case M is real analytic, M is minimal at p if and only if M has finite t y p e at p. T h e m i n i m a l i t y is necessary and sufficient for wedge extendibility. Theorem

6.1. (Tumanov [T1]) Let M be a generic manifold in C N. Suppose M

is minimal at p C M. Then there is a wedge W with edge M near p such that all CR functions on M extend to be holomorphic in W. Theorem

6.2. (Baouendi and Rothschild [BR]) Let M be a generic manifold in

C N. Suppose M is not minimal at p C M. Then for every neighborhood U C M of p there exists a CR function in U that does not extend to any wedge with edge M near p. T h e o r e m s 6.1 and 6.2 generalize Tr@preau's Theorem 1.2. We will give a p r o o f of T h e o r e m 6.1. Let ylp denote the set of all discs A : ~ --~ C N a t t a c h e d to M through p E M, t h a t is A(1) = p. Let ~ denote the inner normal to the unit circle. To every A E ~4p we associate its n o r m a l direction at p: A H [A.(1)] c Np(M), where A~ = OA/Ov, a n d the brackets denote the class of a vector in Np(M). We set F = C o n v {[A~(1)] : A e Ap} C Np(M), the convex span of all Av(1) in Np(M). Lemma

6.3. Suppose that the cone F has nonempty interior. Then all CR func-

tions on M extend to the same wedge W with edge M near p. This l e m m a follows by the edge of the wedge theorem and the following

135

6.4. Let A c AB and ( = [A.(1)] r 0. Then all CR functions on M extend to be CR on a manifold 1~I such that 01VI = M near p and A~(1) E Tp(l~) points inside 1~I. Lemma

Pro@ Let ( z , w ) C C m • C n form a system of coordinates in C N so t h a t w : C TM x C n --~ C n maps T~(M) isomorphically to C n. Let p = 0. T h e n every small disc a t t a c h e d to M is uniquely defined by A(1) and its w-component ~ --* ~(~). By P r o p o s t i o n 3.1, for every/5 = (x0 +iyo, w0) close to p = 0, there is a disc A a t t a c h e d to M through/5, t h a t is A(1) =/5, with w-component ~ ~ @(~) = w(r + w 0 , where ~-* w(~) is the w-component of the given disc A. Let / ~ / = UA((0, 1)) the union of the radii of the discs A. T h e n - g / i s a s m o o t h manifold near p E M . Since 1~/contains A((0, 1)), t h e n A . ( 1 ) E Tp(f/I). Since 1~/ is covered by discs, t h e n all C R functions on M extend to/V/. T h e l e m m a follows. We will show t h a t if M is minimal at p C M , then the cone F has n o n e m p t y interior, and T h e o r e m 6.1 follows from L e m m a 6.3.

7. T h e d e f e c t o f a n a n a l y t i c d i s c We introduce a central notion in the proof of T h e o r e m 6.1, the defect of an analytic disc a t t a c h e d to a C R manifold. The original definition of the defect was given in [T1]. We give a more intuitive geometric definition of the defect due to Baouendi, Rothschild a n d T r @ r e a u [BRT]. Let M be a generic manifold in C N. Let T*I,~ or (1,0) forms in C.g.

Every co E T*I'~

coj C C. Note t h a t T*I'~

N) be the bundle of holomorphic

has the form co = ~cojdzj, where

N) ~ C N • C N is a complex manifold. The conormal

bundle N*(M) of M in C.N is the real dual bundle to N ( M ) . and holomorphic forms by co ~

Reco.

We identify real

In this identification N*(M) becomes a

submanifold in T * I ' ~

N;(M) =

{co e T ; I ' ~

RecolT,(M) = 0}.

Since T* 1,0 ( c N ) is a complex manifold, we can consider analytic discs in it. Let A be an analytic disc a t t a c h e d to M . We say t h a t an analytic disc A* : A --* T * l ' ~ is a lift of A to N* ( M ) if A* (~) E N~(r

N)

for ~ C bA. Let VA be the set of all

lifts of the disc A to N* (M). T h e set VA is a real vector space. We introduce the defect d e f A of t h e disc A as follows. def A = dim VA,

136

VA ----{ A * : A* is a lift of A to N * ( M ) } . Generally, N*(M) is not a C R manifold, so there might not be any nontrivial analytic discs a t t a c h e d to N*(M). Therefore, the existence of nontrivial lifts of A is r e g a r d e d as degeneracy of A, and d e f A measures the degree of this degeneracy. We set

VA(() = {A*(() : A* C VA},

( e bA.

It t u r n s out t h a t the spaces VA(() have the same dimension d e f A . L a m i n a 7.1. For every ( E bA the evaluation map VA --~ VA(() defined as A* ~-*

A*(() is bijective. Proof. Since M is generic, then the fibers Np(M) are t o t a l l y real in T ; I , ~

-~

C N, so t h e complex tangent spaces to N* (M) have dimension at most n -- C R d i m M., Let w : C N --* C n b e a linear projection to a complex subspace t h a t m a p s Tp(M) to C n isomorphically. T h e n t h e complex tangent spaces to N* ( M ) are p r o j e c t e d to C n injectively. By Corollary 3.2 applied to N * ( M ) , every lift A* of A is uniquely defined by t h e b o u n d a r y point A* (() and the w-component of t h e disc A, so the l e m m a follows. We consider the evaluation m a p s ~ - : Ap ~ M,

he(A) = A ( - 1 ) ;

G : Ap --* N p ( M ) ,

G(A) = [A~(1)],

where A~(1) = OA(1)/O~, is the derivative of A in the direction of the inner n o r m a l to bA at 1. By P r o p o s i t i o n 3.1, Ap is identified with a neighborhood of zero in the B a n a c h space of all analytic discsw in C n with w(1) = 0, where n = C R d i m M . T h e n ~" a n d G are s m o o t h maps, so we can consider their derivatives. We set

FA = a a n g e . T ' ( A ) C TA(-1)(M) GA = R a n g e g ' ( A ) C Np(M) M a i n L a m i n a 7.2. ([BRT], [T1]) Let A e Ap. Then the following two statements

hold. (F) FA = VA(--1) • C TA(-1)(M), in particular, d i m F A -- d i m M - d e f A and

T~(_D(M ) C FA. (G) GA = VA(1) • C Np(M), in particular, d i m G A = c o d i m M - d e f A . We first derive T h e o r e m 6.1 from Main L e m m a 7.2.

137

Proof of Theorem 6.1. We will show that there is a small disc A E Ap with d e f A -- 0.

Then by (G), GA -----Np(M). Therefore, F = C o n v (G(Av)) has nonempty interior, and Theorem 6.1 follows by Lemma 6.3. Let d = l i m i n f d e f A as A approaches the constant "disc" {p}. Let Ao E ,4p be a sufficiently small disc with defA0 = d. We can assume that A 0 ( - 1 ) = p, otherwise we replace Ao by the disc ~ H -40(() = A0((2), which has the same defect d and the property 4 0 ( - 1 ) = p. Since all discs A close to Ao have the same defect d, then by (F), the evaluation map ~ has constant rank dim M - d near Ao. Therefore, f maps a neighborhood of A0 to a smooth manifold S C M with dim S -- dim M - d. For every disc A close to A0, we have T ~ ( _ I ) ( M ) C FA = TA(-1)(S). Hence, S is a C R submanifold in M through p with CRdim S -- CRdim M. Since M is minimal, this can happen only if dim S = dim M, so defAo = d -- 0, what we need. The proof is complete.

8. P r o o f o f M a i n L e m m a We begin wth the following factorization lemma (see [BRT]). L e m m a 8.1. Let P be a smooth complex m x m matrix function on bA sufficiently close to the identity I and P(1) = I. Then P admits a decomposition of the f o r m P = R H so that R and H are smooth m • m matrix functions on bA close to I,

R(1) --- H(1) = I, R is real, and H holomorphically extends to A . Proof. The matrix H = R - 1 P is holomorphic and H(1) = I if and only if

(8.1) Let U -- R e ( R - 1 P )

Re(R-1P) = I -TIIm(R-1P). -- R - 1 R e P .

Since P is close to I, then R e P is invertible,

R -1 -- U ( R e P ) -1, and (8.1) turns into

(8.2)

U = I - TI(U(ReP)-IImP).

Since Im P is small, this equation has a unique solution U, which is real, close to I, and U(1) -- I. Then R :-- ( R e P ) U -1 is real, and (8.2) implies (8.1). Hence, H -- R - 1 p holomorphically extends to A. Lemma is proved. Let M be a generic manifold in C N defined by a local equation T ( z ) = o,

138

where r is a smooth R m valued function of Z = (z, w) E C m x C n = C N. We assume 0 C M and rz (0, 0) = I,

r~ (0, 0) = 0,

where rz = Or/Oz, r~ = Or/Ow. We denote by (w,~) the value o f t e T ; I ' ~

N) on ~ 9 Tp(CN). We then have

Tp(M) = {~ 9 Tp(cN) : Re (Or(p),() = 0}, Tp(M) = {~ 9 Tp(CN) : (Or(p),~) = 0}. In terms of the defining function r, the conformal bundle N*(M) is described by

Np(M) = {)~Or(p) : A 9 Rm}, where A is a row vector and r a column vector. Recall that Re (w, ~} defines the duality between N* (M) and N(M). Recall also that the operator of complex structure J gives an isomorphism T(M)/TC(M) ~ N(M) by [~] ~-* [J~]. Therefore Im (w, ~) defines duality between N*(M) and T(M)/TC(M).

Proof of (F) in Main Lemma. Since T(M) is orthogonal to N* (M) with respect to Re (., .), we mean by VA(--1) • the orthogonal complement with respect to (., .} or

Ira(., .). We first describe FA = Range~-'(A). The tangent space TA(AB) consists of all analytic dics A in T ( C N) attached to T(M) such that ~i(1) = 0 and for every ( 9 bA we have A(() 9 TA(r

that is Re

(Or(A(~)), A(r

= 0,

~ 9 bA.

Since ~ is an evaluation map, we have H ( A ) A = A ( - 1 ) and FA = { A ( - 1 ) : A 9 TA(.Ap)}. Since T(M) is orthogonal to N* (M) with respect to Re (., .), then for every A* 9

VA and every A 9 TA(Ap), we have R e ( A * , A ) = 0. The function (A*,A) is holomorphic in A, so it is constant.

Since A(1) = 0, we have ( A * , A )

-

O.

In

particular, ( A * ( - 1 ) , A ( - 1 ) ) -- 0. Thus FA C VA(--1) • To prove that FA D VA(--1) • T ~ ( _ I ) ( M ) such that r

we will show that for every real form r 9

= 0, there exists A* 9 VA such that r = I m A * ( - 1 ) .

We denote B(r

= rz(A(r

= (rz(A(r

r~(A(r

= (P(r

Q(r

r c bA.

139

We represent vectors by columns and covectors by rows. T h e n for A E TA(.Ap) we have Re (BA]bA) = 0. Likewise, A* E V A if and only if A*[bA = AB, where A is a •m-valued function such t h a t AB holomorphically extends to A. Let P = R H be the decomposition provided by L e m m a 8.1. T h e n by the substitutions

B ~ R-1B

(8.3)

(., 0i) 0

,

A ---* AR,

A ---* H A ,

we reduce the question to the case in which P - I , t h a t is B -- (I, Q). T h e n the condition Re ( B A ) = 0 for A = (s w) turns into

(8.4)

Re

+ Q(r162

= 0,

r

To satisfy this condition, we take an a r b i t r a r y analytic disc zb : A -~ C n with ~b(1) = 0 a n d define s

by

(8.5)

~ = - ( R e (Qzb) + iT1Re (Q(v)).

T h e real form r is defined for all Z = (z, w) E C N such t h a t Re ( B Z ) = Re (z + Q ( - 1 ) w ) = 0. T h e n r must have the form = Re (aw) - iA(z + Q ( - 1 ) w ) ,

r

where a E C n a n d A c R m are constant rows. T h e condition r

-- 0 takes

the form

(8.6)

Re (a~b(-1)) - iA(~(-1) + Q(-1)~b(-1)) = 0.

F i r s t restrict our a t t e n t i o n to A for which ~b(-1) ---- 0. As

T h e n by (8.6) we have

= 0, whence by (8.5)

(8.7)

(T1Re ( A Q w ) ) ( - 1 ) = O.

Note t h a t

(8.8)

2 f~

(Tlf)(-1) = ~

f(() d(

ebA ~ : - - 1 "

T h e condition (8.7) holds for all holomorphic zb with zb(=kl) = 0. T h e n by (8.8) the function AQ is orthogonal to all holomorphic functions in A therefore AQ extends to A holomorphically.

140

Now regardless of ~b(-1), we have TIRe (AQ~b) -- Im (AQ@), and (8.5) yields ABA = A(~ + Q@) - 0. In particular, ,~(~(-1) + Q ( - 1 ) ~ b ( - 1 ) ) = 0, and by (8.6) we have Re(c~@(-1)) -- 0. Since ~b(-1) is arbitrary, then a = 0 and r Im ( A B ( - 1 ) Z ) .

Since AB = (A, AQ) is holomorphic, then it represents A* E

--

VA.

Thus r ----I m A * ( - 1 ) , what we need. The proof is complete.

Proof of ((7) in Main Lemma.

The orthogonal complement in the statement of

N*(M)

(G) refers to the duality between

ip(U),

G'(A)ii = [iiv(1)] E

N(M)

given by Re (., .). We have

so

GA = a a n g e ~ ' ( A ) Recall that for every A* E

and

VA

= {[Av(1)] : A E

and every iil E

TA(.Ap), we

(A*(1),A~(1)) --

Differentiating this relation yields

TA(.Ap)}. have

(A*,A) --

0.

0 because A(1) = 0. Thus

GA C VA(1) • We now prove that

GAD

VA(1) •

that is G~ C VA(1). Let w E G~ C

N~(M).

The form w is represented by AB(1), where )~ E R m. The condition w E GA ~ means that for every A E

TA(Ap),

we have

Re(AB(1)A.(1))=

0. We need to show that

there exists A* EVA such that A*(1) = AB(1). By the substitution (8.3), we reduce the question to the case in which P - I, that is B -- ( I , Q ) .

(~, (v) E TA(.Ap), we

Note that B(1) --- ( I , Q ( 1 ) ) -- (I, 0). Then for every A = have

(8.9)

Re (A(~(1)) -- 0,

where ~ = c9~/0~. For every smooth harmonic function f in A with f(1) = 0 we have

(8.10)

Of(l) _ 1 ,/~o2~

f!eiO)dO_

i~

f(~)d~

We restrict to A for which @ = O(l~ - 112). Then (8.9) turns into Re

jo 2~ )~(e i~ dO

T~-I- F =0,

and by (8.4) and (8.10) we get Im ~ Eba

AQ(r162 de (( _ 1) 2 -- 0.

This relation implies that AQ is orthogonal to all holomorphic functions in A, hence AQ extends holomorphicMly to A. Now AB = (/k, AQ) defines A* EVA with A*(1) = AB(1), what we need. The proof is complete.

141

References

[AH]

Ayrapetian, R. A. and Henkin, G. M., Analytic continuation of CR functions through the "edge of the wedge", Dokl. Akad. Nauk SSSR 259 (1981), 777-781.

[BR]

Baouendi, M. S. and Rothschild, L., Cauchy-Riemann functions on manifolds of higher codimension in complex space, Invent. Math. 101 (1990), 45-56.

[BRT] Baouendi, M. S., Rothschild, L. and Tr6preau, J.-M., On the geometry of analytic discs attached to real manifolds, J. Diff. Geom. 39 (1994), 379-405. [BT]

Baouendi, M. S. and Treves, F., A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. 114 (1981), 387-421.

[Bi]

Bishop, E., Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1-21.

[Bo]

Boggess, A., CR manifolds and the tangential Cauchy-Riemann complex, CRC Press, 1991.

[Tr]

Tr6preau, J.-M., Sur le prolongement holomo~he des fonctions CR ddfinies sur une hypersurface reUe de classe C 2 dans C n, C. R. Acad. Sci. Paris S~r. I Math. 301 (1985, no. 3), 61-63.

[T1]

Tumanov, A. E., Extending CR functions on a manifold of finite type over a wedge, Mat. Sbornik 136 (1988), 129-140.

IT2]

__, On the propagation of extendibility of CR functions, In: V. Ancona et al.: Complex analysis and geometry. (Lect. Notes Pure Appl. Math. vol. 173) Basel, New York: Maxcel-Dekker, 1995.

DEPARTMENT OF MATHEMATICS UNIVERSITY OF ILLINOIS 1409 WEST GREEN STREET URBANA,ILLINOIS 61801 USA

C.I.M.E. S e s s i o n on " I n t e g r a l G e o m e t r y , R a d o n T r a n s f o r m s and C o m p l e x A n a l y s i s " List of Participants

A. ABOELAZ, Dept. of Math. and lnfo., Casablanca, Marocco F. ANDREANO, Via Cagliari 11, 00198 Roma, Italy F. ASTENGO, Dip.to di Mat. del Politecnico, Corso Duca degli Abruzzi 24, 10129 Torino, Italy L. ATANASI, Dip.to di Mat.,Univ., P.le A. Moro 5, 00185 Roma, Italy M.P. BERNARDI, Dip.to di Mat., Univ., Via Abbiategrasso 209, 27100 Pavia, Italy C. BOITI, Dip.to di Mat., Univ., Via Buonarroti 2, 56127 Pisa, Italy C. BONDIOLI, Dip.to di Mat., Univ. Via Abbiategrasso 209, 27100 Pavia, Italy L. BRANDOLINI, Dip.to di Mat., Univ., Via Saldini 50. 20133 Milano, Italy L. CAPOGNA, Dept. of Math., Purdue Univ., 1395 Mathematical sciences Bldg, West Lafayette, IN, USA L. CARBONE, Dept. of Math., Columbia Univ., New York, NY 10027, USA G. CARCANO, Dip.to di Metodi Quantitativi, Contrada S. Chiara 48/B, 25122 Brescia, Italy V. CASARINO, Dip.to di Mat., Univ., Corso Carlo Alberto 10, 10123 Torino, Italy P. CIA'I"rI, Dip.to di Mat. del Politecnico, Corso Duca degli Abruzzi 24. 10129 Torino, Italy S. COEN, Dip.to di Matematica, Univ., Piazza di Porta S. Donato 5, 40127 Bologna, Italy S. CUCCAGNA, Dept. of Math., Princeton Univ., 304 Fine Hall, Princeton, NJ 08544-1000, USA A. D'AGNOLO, Dip.to di Mat., Univ., Via Belzoni 7, 35131 Padova, Italy R. DAHER, Univ. Hassan II, Fac. des Sciences de Ain Chok, Casablanca, Maroeco C. DE FABRITIIS, SISSA, Via Beirut 2-4, 34013 Trieste, Italy G. DIN'I, Dip.to di Mat., Univ., Viale Morgagni 67/a, 50134 Firenze, Italy G. FELS, Univ. of Essen, Robert Kock Str. 1, Witten, Germany P. GRACZYK, D~pt. de Math., Univ. d'Angers, 2 bd. Lavoisier, 49045 Angers, Prance D. IACOVIELLO, Dip.to di lnfo. e Sist., Univ. "La Sapienza", 00185 Roma, Italy B. IVARSSON, Matematiska Institutionen, Box 480, 751 06 Uppsala, Sweden O. OKTEN, Dept. of Math., Stockholm Univ., 10691 Stockholm, Sweden M. V. OULD MOUSTAPHA, ICTP Maths Section, PO Box 586, 34100 Trieste, Italy C. PERELLI CIPPO, Dip.to di Mat. del Politecnico, Piazza L. da Vinci, 323, 20133 Milano, Italy A. PEROTYI, Dip.to di Mat., Univ., Via Saldini 50, 20133 Miolano, Italy M. PEVSNER, Inst. de Math., Univ. P. et M. Curie, 4 pl. Jussieu, 75252 Paris, Prance P. ROSSI, Dip.to di Mat., Univ. di Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy F. ROUVIERE, Univ. de Nice, Lab. Dieudonne, Parc Valrose, 06108 Nice Cedex 2, France D. ROUX, Dip.to di Mat., Univ., Via Saldini 50, 20133 Milano, Italy B. ROUBIN, Dept. of Math., Givat Ram,m The Ebrew Univ., 91904 Jerusalem, Israel E. SELIVANOVA, Rokossovskogo 17-113, Nizhni Novgorod 603136, Russia A. SELVAGGI PRIMICERIO, Dip.to di Mat., Univ., Viale Morgagni 67/a, 50134 Firenze, Italy T. KOBAYASHI, Graduate School of Math. Sciences, Univ. of Tokyo, Komaba, Meguro, Tokyo 153, Japan A. KORANYI, Math. Dept., Lehman College, Bronx 10468, USA L. LANZANI, Dept. of Math., Purdue Univ., 1395 Mathematical Sciences Bldg, West Lafayette, IN, USA H.-Q- LI, Dept. de Math., Univ. de Paris-Sud, 91405 Orsay, France R. MANFRIN, Scuola Normale Superiore, Piazza dei Cavalieri 7, 57126 Pisa, Italy C. MARASTONI, Dip.to di Mat. pura ed appl., Univ., Via Belzoni 7, 35131 Padova E. MIHAILESKU, Bd. Camil Ressu 27, Bloc NI, Sc.4m Ap183, Sector 3, Bucuresti, Romania C. MEDORI, SISSA, Via Beirut 2-4, 34013 Trieste, Italy F. MEYLAN, $bb Cigale, 1010 Lausanne, Switzerland C. MIRAN, Dept. of Math., Univ. of Ljubljana, Jadranska 19, 1 111 Ljubljana, Slovenia S. OFMAN, Inst. de Math., Grom. et Dynam., Univ. Paris 7, 2 place Jussieu, 75251 Paris, France L. SKRZYPCZAK, Fac. of Math. and Comp. Sci., A. Mickiewicz Univ., Matejki 48/49, 60769 Poznan, Poland M. SUNDARI, ICTP, PO Box 586, 34100 Trieste, Italy F. TONIN, Inst. de Math., Univ. P. et M. Curie, 4 pl. Jussieu, 75252 Paris, France S. TRAPANI, Dip.to di Mat., Univ. di Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy G. TRAVAGLINI, Dip.to di Mat., Univ., Via Saldini 50, 20133 Milano, Italy S. VENTURINL Dip.to di Mat., Univ., Piazza di Porta S. Donato 5, 40127 Bologna, Italy G. WEILL. Polytechnic Univ., 6 Metrotech Ctr. i, Brooklyn, NY 11201, USA A. ZAPPA. Dip.to di Mat., Univ., Via Dodecaneso 35, 16146 Genova, Italy

144

L I S T O F C.I.M.E.

1954

-

SEMINARS

1. Analisi

funzionale

2. Quadratura 3. Equazioni

1955

-

Publisher

4. Teorema

delle

C.I.M.E.

superficie

differenziali

di R i e m a n n - R o c h

non

e questioni

connesse

lineari

e questioni

connesse

5. Teoria dei numeri 6. Topologia 7. Teorie non linearizzate

in e l a s t i c i t i ,

idrodinamica,aerodinamica

8. Geometria p r o i e t t i v o - d i f f e r e n z i a l e

1956

-

9. Equazioni i0.

alle derivate

Propagazione

delle onde

ii. Teoria della funzioni funzioni

1957

-

-

aritmetica

13.

singolari

Integrali

e problemi

16. Problemi 17.

elettromagnetiche complesse

e delle

e algebrica e questioni (2 vol.)

attuali

di g e o m e t r i a

Ii principio

(2 vol.) connesse

in r e l a t i v i t &

differenziale

di m i n i m o

generale

in grande

e le sue a p p l i c a z i o n i

alle equazioni

funzionali

1959

- 18.

Induzione

e statistica

19. Teoria algebrica 20. Gruppi,

1960

- 21. 22.

1961

Sistemi

dei m e c c a n i s m i

anelli di Lie e t e o r i a

dinamici

e teoremi

Forme differenziali

- 23. Geometria 24.

e loro

del calcolo

delle

Teoria delle distribuzioni

25. Onde superficiali

1962

reali

automorfe

12. Geometria

15. V e d u t e

a caratteristiche

di p i ~ v a r i a b i l i

14. Teoria della turbolenza

1958

parziali

- 26. Topologia 27. Autovalori

differenziale e autosoluzioni

28. M a g n e t o f l u i d o d i n a m i c a

automatici della

(2 vol.)

coomologia

ergodici integrali

variazioni

(2 vol.)

145 1963

1964

- 29. Equazioni 30.

Funzioni

31.

Propriet~

- 32.

differenziali e variet~

di media e t e o r e m i

Relativit~

33. Dinamica

- 36. Non-linear

- 39.

di a n a l i s i

differenziali

continuum

non

numerica lineari

theories

Some aspects of ring t h e o r y

38. Mathematical

1966

in Fisica M a t e m a t i c a

generale

35. Equazioni

37.

di c o n f r o n t o

dei gas rarefatti

34. Alctule questioni

1965

astratte

complesse

Calculus

optimization

in e c o n o m i c s

of variations

Ed. Cremonese,

40. Economia matematica 41.

Classi

caratteristiche

42. Some aspects

1967

- 43. M o d e r n

questions

44. Numerical 45. G e o m e t r y

1968

- 46. 47.

1969

- 49.

analysis

of p a r t i a l

mechanics differential

bounded

equations

domains

and o b s e r v a b i l i t y

Pseudo-differential

operators

of mathematical

Potential

connesse

theory

of c e l e s t i a l

of h o m o g e n e o u s

Controllability

48. Aspects

e questioni

of d i f f u s i o n

logic

theory

50. Non-linear

continuum

theories

in m e c h a n i c s

a n d physics

and their applications 51. Questions

1970

of a l g e b r a i c

- 52. Relativistic

varieties

fluid d y n a m i c s

53. T h e o r y of group r e p r e s e n t a t i o n s

1971

54.

Functional

55.

Problems

- 56.

equations

and Fourier

analysis

and i n e q u a l i t i e s

in n o n - l i n e a r

analysis

Stereodynamics

57.

Constructive

58.

Categories

aspects

of f u n c t i o n a l

and c o m m u t a t i v e

algebra

analysis

(2 vol. )

Firenze

146 1972 - 59. Non-linear mechanics 60. Finite geometric structures and t h e i r applications 61. Geometric measure theory and m i n i m a l

1973

surfaces

- 62. Complex analysis 63. New variational techniques

in m a t h e m a t i c a l physics

64. Spectral analysis

1974 - 65. Stability problems 66. Singularities of analytic spaces 67. Eigenvalues of non linear p r o b l e m s

1975

~ 68. Theoretical computer sciences 69. Model theory and a p p l i c a t i o n s 70. Differential operators and m a n i f o l d s

Ed Li_uuori,

1976 - 71. Statistical Mechanics

Napoli

72. Hyperbolicity 73. Differential topology

1977 - 74. Materials with m e m o r y 75. Pseudodifferential operators w i t h a p p l i c a t i o n s 76. Algebraic surfaces

1978 - 77. Stochastic differential e q u a t i o n s 78. Dynamical systems

Ed Liguori,

1979 - 79. Recursion theory and c o m p u t a t i o n a l

Napoli and B i r h ~ u s e r V e r l a g

complexity

80. Mathematics of b i o l o g y

1980

-

81.

Wave

propagation

82. Harmonic analysis and group r e p r e s e n t a t i o n s 83. Matroid theory and its a p p l i c a t i o n s

1981 - 84. Kinetic Theories and the B o l t z m a n n E q u a t i o n

(LI~M 1048) S p r i n g e r - V e r l a g

85. Algebraic Threefolds

(I/qM 947)

86. Nonlinear Filtering and S t o c h a s t i c Control

(LNM

972)

(LNM

996)

1982 ~ 87. Invariant Theory 88. Thermodynamics and C o n s t i t u t i v e E q u a t i o n s

(LN Physics 228)

89. Fluid Dynamics

(LNM 1047)

147

1983

90.

-

Complete

Intersections

91. Bifurcation 92. Numerical

1984

- 93. Harmonic

Methods

Mappings

94. Schr6dinger 95. Buildings

1985

- 96. 97.

(LNM 1092)

T h e o r y and A p p l i c a t i o n s in Fluid D y n a m i c s

(LNM 1127)

and M i n i m a l

(I.,NM 1161)

Immersions

and the G e o m e t r y

of D i a g r a m s

and Analysis

Some Problems

in N o n l i n e a r

Diffusion

98. T h e o r y of Moduli

1986

-

99.

Inverse

i01.

1987

- 102.

- 104. 105.

1989

1990

- 108.

"

(LNM 1224)

"

(LNM 1225)

Economics

(LNM 1330)

Optimization

(LNM 1403)

Relativistic

Fluid D y n a m i c s

(LNM 1385)

in Calculus

Logic and Computer Global

- 106. Methods 107.

(LNM 1206)

Combinatorial

103. Topics

1988

(LNMlZ81)

(LNM 1337)

Problems

i00. Mathematical

Geometry

of V a r i a t i o n s

Geometric

Analysis

Topology:

109.

H

Control

ll0.

Mathematical

(LNM 1365)

Science

(LNM 1429)

and M a t h e m a t i c a l

of n o n c o n v e x

Microlocal

Physics

analysis

(LNM 1451)

(LNM 1446)

and A p p l i c a t i o n s

(LNM 1495)

Recent

(LNM 1504)

Developments

Theory

(LNM 1496)

Modelling

of I n d u s t r i a l

(LNM 1521)

Processes

1991

- IIi.

Topological

Methods

Differential

Algebraic

113.

to Chaos

Transition

(LNM 1537)

for O r d i n a r y

Equations

112. Arithmetic

Geometry in C l a s s i c a l

(LNM 1553) and

(LNM 1589)

Quantum Mechanics

i992

- 114. i15.

Dirichlet DLModules,

Forms

(LNM 1563)

Representation

Theory,

(LNM i565)

and Quantum Groups 116. N o n e q u i l i b r i u m Systems

"

(.T.,NM 1159)

Operators

Probability

Springer-Verlag

(LNM 1057)

Problems

in M a n y - P a r t i c l e

(LNM 1551)

148 1993 - 117. Integrable Systems and Quantum Groups

1994

-

(LNM 1620)

118. Algebraic Cycles and Hodge Theory

(LNM 1594)

119. Phase Transitions and Hysteresis

(LNM 1584)

120. Recent Mathematical Methods in

(LNM 1640)

Nonlinear Wave Propagation 121. Dynamical Systems

(LNM 1609)

122. Transcendental Methods in Algebraic

(LNM 1646)

Geometry

1995

-

123. Probabilistic Models for Nonlinear PDE's

(LNM 1627)

124. Viscosity Solutions and Applications

(LNM 1660)

125. Vector Bundles on Curves. New Directions

(LNM 1649)

1996 - 126. Integral Geometry, Radon Transforms

(LNM 1684)

and Complex Analysis 127. Calculus of Variations and Geometric

to appear

Evolution Problems 128. Financial Mathematics

1997 - 129. Mathematics Inspired by Biology 130. Advanced Numerical Approximation of

(LNM 1656)

to appear to appear

Nonlinear Hyperbolic Equations 131. Arithmetic Theory of Elliptic Curves

to appear

132. Quantum Cohomology

to appear

Springer-Verlag

F O N D A Z I O N E C.I.M.E. C E N T R O INTERNAZIONALE M A T E M A T I C O ESTIVO INTERNATIONAL M A T H E M A T I C A L S U M M E R CENTER

"Mathematics Inspired by Biology" is the subject of the First 1997 C.I.M.E. Session. The session, sponsored by the Consiglio Nazionale delle Ricerche (C.N.R), the Ministero dell'Universit~ e della Ricerca Scientifica 9 Tecnologica (IVI.U.R.S.T.),and European Community will take place, under the scientific directions of Professors VINCENZD CAPASSO (Universitb di Milano) and ODO DIEKMANN (University of Uu~cht) in Manina Franca (Taranto), from 13 to 20 June, 1997.

Courses

a)

Dyn=mlcs of physiologically structured populations (6 lectures in English) Prof. Odo DIEKMANN (University of Utrecht)

Outline:

I. Formulation and analysis of general linear models. The connection with multi-type branching processes. The definition of the basic reproduction ration R o and the Malthusian parameter r. Spectral analysis and asymptotic large time behaviou~. 2. Nonlinear models: density dependence through feedback via environmental interaction variables. Stability boundaries in parameter space. Numerical bifurcation methods. 3. Evolutionary considerations. Invasibility and ESS (Evolutionarily Stable Strategies). Relationship with fitness meal.ires.

4. Case studies:

cannibalism Daplmia feeding on algae - reproduction strategy of Salmon

-

References I.A.J. Melz & O. Diekmann (eds.),Dynamics of Physiologically Structured Populations,Lect.Notes in Biomath. 68, 19860 Springer. - O. Diekmann0 M. Gyllenberg0 1. A. J. Metz & H. R. Thieme, On the formulation and analysis of general deterministicstructuredpopulationmodels, L Linear theory,preprinL P. Jagers,The growth and stabilizationof populations, StatisticalScience 6 (I991 ),269-283. P. lagers,The deterministicevolutionof general branching populations, preprinl. O. Diekmann, M. A. Kirkilonis,B. Lisser, M. Louter-Nool, A. M. de Roos & B. P. Sommeijer, Numerical continuation of equilibriaof physiologicallystructuredpopulation models, preprint. S. D. Mylius & O. Diekmann, On evolutionarilystable lifehistories,optimization and the need to be specificabout density dependence, OIKOS 74 (1995),218-224. 1. A. J.Meetz, S. M. Mylius & O. Diekmann, W h e n does evolution opdmise? preprinL A. M. de Roos, A gende in~oduction to physiologicallystructuredpopulation models, preprinL F. van den Bosch, A. M. de Roos & W. Gabriel, Cannibalism as a lifeboat mechanism, 1. Math. Biol. 26 (1988), 619-633. V. Kaitala & W. W. Getz, Population dynamics and harvesting of semel-parous species with phenotypic and genotypic variabilityin reproductiveage, I. Math. Biol. 33 (1995), 521-556.

150 b)

When is space important in modelling biological systems? (6 lectures in n.,nglish). Prof. Rick DURRETI" (Comell University)

Outline: In these lectures I will give an introduction to stochastic spatial models (also called cellular automaLa or interacting particle systems) and relate their properties to those of the ordinary differential equations that result if one ignores space and assumes instead that all individuals interact equally. In brief one finds that if the ODE has an ~ n g fixed point then both approaches (spatial and non-spatial) agree that coexistence will occur. How if the ODE has two or more locally stable equilibria or periodic orbits? Then the two approaches can come to radically different predictions. We will illustrate these principles by the study of a number of different examples - competitionof Daphnia species in rock pools - allelopathy in E. coli spatial versions of Prisoner's dilemma in Maynard Smith's evolutionary games framework - a species competition model of Silvertown et al. and how it contrasts with a three species ODE system of May and Leonard -

References

-

c)

17,.DurreU (1955) Ten lectures on Particle Systems, pages 97-201 in Springer Lecture Notes in Math. 1608. R. Durrett and S. Levin (1994) Stochastic spatial models: a user's guide to ecological applications. Phil. Trans. Roy. Soc. B 343,329-350. R. Durrett and S. Levin (1994) The importance of being discrete and spatial. Theor. Pop. Biol. 46 (1994), 363-394. R. Durrett and S. Levin (1996) Allelopathy in spatially distributed populations. Preprint R. Durrett and C. Neuhauser (1996) Coexistence results for some competition models. Random walk systems modeling spread and interaction (6 lectures in English). Prof. K. P. HADh-:IFR (University of Tiibingen)

Outline: Random walk systems are semilinear systems of hyperbolic equations that describe spatial spread and interaction of species. From a modeling point of view they are similar to reaction diffusion equations but they do not show infinitely fast propagation. These systems comprise correlated random walks, certain types of Boltzmann equations, the Cattaneo system, and others. Mathematically, they are closely related to damped wave equations. The aim of the course is the derivation of such systems from biological modeling assumptions, exploration of the connections to other approachas to spatial spread (parabolic equations, stochastic processes), application to biological problems and presentation of a qualitative theory, as far as it is available 1. The problems and their history Reaction diffusion equations; velocity jump processes and Boltzmann type equations; correlated random walks; nonlinear interactions; Cattaneo problems; reaction telegraph equations; boundary conditions. 2. Linear theory Explicit representations; operator semigroup theory; positivity properties; spectral properties; compactness properties. 3. Semilinear systems Invarianee and monotonicity; Lyapunov functions; gradient systems; stationary points; global attractors. 4. Spatial spread Travelling fronts and pulses; spread of epidemics; bifurcation and patterns; Turing phenomenon; free boundary value problems. 5. Comparison of hyperbolic and parabolic problems.

References Goldstein, S, On diffusion by discontinuous movements and the telegraph equation. Quart. J. Mech. Appl. Math. 4 (1951), 129-156. Kac, M., A stochastic model related to the telegrapher's equation. (1956), reprinted Rocky Math. J. 4 (1974), 497509. Dunbar, S., A branching random evolution and a nonlinear hyperbolic equation. SIAM J. Appl. Math. 48 (1988), 1510-1526. Othmer, H. G., Dunbar, S., AIr, W., Models of dispersal in biological systems. J. Math. Biol. 26 (1988), 263-298. Hale, J. K., Asymptotic behavior of dissipative systems. Amer. Math. Soc., Providence, R.I. 1988. Temam, R., Infinite dimensional dynamical systems in mechanics and physics. Appl. Math. Series 68~ Springer 1988.

151 Hadeler, K. P., Reaction telegraph equations and random walks. Canad. Appl. Math. Quart. 2 (1994), 27-43. Hadeler, K. P., Reaction telegraph equations and random walk systems. 31 pp. In: S. van Suien, S. Verdoyn Ltmel (ods.), "Dynamical systems and their applications in science". Royal Academy of the Netherlands, North Holland 1966. d)

Mathematical Modelling in Morphogenesis (6 lectures in English). Prof. Philip MAINI (Oxford University)

Outline: A cenual issue in developmental biology is the formation of spatial and spatio temporal panems in the early embryon, a process k~own as morpboganesis. In 1952, Turing published a seminal paper in which he showed that a system of chemicals, reacting and diffusing, could spontaneously generate spatial patterns in chemical concentrations, and he termed this process "diffusion-driven instability". This paper has generated a great deal of research into more general reaction diffusion systems, consisting of nonlinear coupled parabolic partial differential equations (Turing's model was linear), both from the viewpoint of mathematical theory, and from the application to diverse patterning phenomena in development. In these lectures I shall discuss the analysis of these equations and show how this analysis carries over to other models, for example, mechanocbemical models. Specifically, I shall focus on linear analysis and nonlinear bifurcation analysis. This analysis will be extended to non-standard problems.

References

e)

G. C. Cruywagen, P.IC Maini, J. D. Murray, Sequential pattern formation in a model for skin morphogenesis, IMA J. Math. Appl. Meal. & Biol., 9 (1992), 22"/-248. R. Dillon, P.K. Maini, H. G. Othrner, Pattern formation in generalised Turing systems. I. Steady-slate patterns in systems with mixed boundary conditions, J. Math. Biol., 32 (1994), 345-393. P. K. Maloi, D. L. Benson, J. A. Shcrratr, Pattern formation in reaction diffusion models with spatially inhomogencous diffusion coefficients, IMA J. Math. Appl. Med. & Biol., ) (1992), 197-213. J. D. Murray, Mathematical biology, Springer Verlag, 1989. A.M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. Lond., B 237 (1952), 37-72. The Dynlmlcs of Competition (6 lectures in English). Prof. Hal L. SMITH (Arizona Slate University, Tempe)

Outline: Competitive relations among populations of organisms are among the most studied by ecologists and there is a vast literature on mathematical modeling of competition. Quite recently, there have been a number of breakthroughs in the mathematical understanding of the dynamics of competitive systems, that is, of those systems of ordinary, delay, difference and partial differential equations arising in the modeling of competition. In these lectures I intend to describe some of the new results and consider their applications to microbial competition for nutrients in a chemostat.

References deMouoni, P. and Schiaffino, A., Competition systems with periodic coefficients: a geometric approach, J. Math. Biology, II (1982), 319-335. Hess, P. and Lazar, A. C., On an abstract competition model and applications, Nonlinear Analysis T.M.A., 16 (1991), 917-940. Hess, P., Periodic-parabolic boundary value problems and positivity, Longman Scientific and Technical, New York, 1991. Hirsch, M., Systems of differential equations that are competitive or cooperative. VI: A local C r closing lemma for 3-dimensional systems, Ergod. Th. Dynamical Sys., 11 (1991), 443-454. Smith, H. L., Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Math. Surveys and Monographs, 41, Amer. Math. Soc. 1995. Smith, H. L., A discrete, size-sWactured model of microbial growth and competition in the chemostat, to appear, J. Math. Biology. Smith, H. L. and Waltman, P., The theo~' of the chemostat, dynamics of microbial competition, Cambridge Studies in Math. Biology, Cambridge University Press, 1995. Hsu, S.-B., Smith, H. L. and Waltman, P., Competitive exclusion and coexistence for competitive systems in ordered Banach spaces, to appear, Trans. Amer. Math. Soc. Hsu, S. B., Smith, H. L. and Waltmanm P., Dynamics of Competition in the onstirred chemostat, Canadian Applied Math. Quart., 2 (1994), 461-483.

FONDAZIONE C.I.M.E. CENTRO INTERNAZIONALE MATEMATICO ESTIVO INTERNATIONAL MATHEMATICAL SUMMER CENTER "Advanced Numerical Approximation of Nonlinear Hyperbolic Equations" is the subjectof the Second 1997 C.I.M.E. Session.

The session, sponsored by the Consiglio Nazionale delle Ricerehe (C.N.R), the Ministero dell'Universiti 9 della Ricerca Scientifica 9 Tecnologice (M.U.R.S.T.), and European Community wilt take place, under the scientific direction of Professor ALFIO QUARTERONI (Politecnico di Milano) at Grand Hotel San Michele, Cewaro (Cosenza), from 23 to 28 June, 1997.

Courses

a)

Discontinuous Galerkin Methods for Nonlinear Conservation Laws (5 lectures in English). Prof. Bemardo COCKBURN (University of Minnesota, Minneapolis)

Outline: Lecture 1: The nonlinear scalar conservation law as paradigm. The continuous dependence slruct'ure associated with the vanishing viscosity method for nonlinear scalar conservation laws. The enU'opy inequality and the enU'opy solution. The DG method as a vanishing viscosity method.

l..r~mre 2: Feam~s of the IX] method. The use of two-point monotone numerical fluxes as an artificial viscosity associated with intexr edges: Relation with monotone schemes. Slope limiting as an arl.ificial viscosity associate with the interior of the elements: Relation with the streamline-diffusion method. Lecture 3: Theoreticalpropertiesof the D G method. Stabilitypropertiesand a prioriand a posteriorierrorestimates. Lecture 4: Computational results for the scalar conservation law. Lecture 5: Extension to general multidimensional hyperbolic systems and computational results for the Euler equations of gas dynamics. b)

Adaptive methods for differential equations and application to compressible flow problems (5 lectures in English) Prof. Clacs JOHNSON (Chalmers University of Technology, Gtteborg)

Outline: We present a general methodology for adaptive error cona'o] for Galerldn methods for differential equations based on a posteriori error estimates involving the residual of the computed solution. The methodology is developed in the monograph Computational Differential Equations by Eriksson, Estep, Hansbo and Johnson, Cambridge University Press, 1996, and the companion volume Advanced Computational Differential Equations, by Eriksson, Esu~p, Hansbo, Johnson and Levenstarn (in preparation), and is realized in the software fem]ab available on Internet h t t : p : / / w ' , r ~ . m a t h . c h a l m e r s , s e / f e m l a b . The a posteriori error estimates involve stability factors which are

153 estimated through auxiliary computation solving dual linearized problems. The size of the stability factor determines if the the specific problem considered is computable in the sense that the error in some norm can be made sufficiently small with the available computational resources. We present applications to a variety of problems including heat conduction, compressible and incompressible fluid flow, reaction-diffusion problems, wave propagation, elasticity/plasticity, and systems of ordinary differential equations. Lecture 1: Introduction. A posteriori error estimates for Galerkin methods Lecture 2: Applications to diffusion problems Lecture 3: Applications to compressible and incompressible flow Lecture 4: Applications to reaction-diffusion problems Lecture 5: Applications to wave propagation, and elasticity/plasticity. l.,eeture 6: Applications to dynamical systems. c)

Essentially Non-Oscillatory (ENO) And Weighted, Essentially Non-Oscillatory (WENO) Schemes For Hyperbolic Conservation Laws (6 lectures in English) Prof. Chi-Wang SHU (Brown University, Providence)

Outline:

In this mini-course we will describe the constrdction,analysis, and application of ENO (EssentiallyNon-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws. ENO and WENO schemes are high order accurate finitedifference schemes designed for problems with piecewisc smooth solutions with discontinuities.The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the smoothest local stencil,hence avoiding crossing discontinuitiesin the interpolationas much as possible.The talkwill be basicallyself-contained,assuming only the background of hyperbolic conservation laws which will be provided by the fu'stlecturesof Professor Tadmor. Lecture I: ENO and WENO interpolation We ",viiidescribe the basic idea of ENO interpolation,sta,"dng from the Newton form of one dimensional polynomial interpolations.We will show how the local smoothness of a function can be effectivelyrepresented by its divided or undivided differences,and how this information can help in choosing the stencilsin ENO or the weights in W'ENO. Approximation resultswill be presented. Lecture 2: More on interpolation It -,rillbe a continuation of the fu'st.We will discuss some advanced topics in the ENO and WENO interpolation,such as differentbuilding blocks, multi dimensions including general triangulations,sub-callresolutions, etc. Lecture 3: Two formulations of schemes for conservation laws We will describe the conservative formulations for numerical schemes approximating a scalar,one dimensional conservation law. Both the cell averaged (finite volume) and point value (finitedifference) formulations will be provided, and theu" similarities,differences,and relativeadvantages will be discussed. ENO and WENO interpolation procedure developed in the firsttwo lectures will then be applied to both formulations. Total variation stable time discretization will be discussed. Lecture 4: Two dimensions and systems Is a continuation of the third. We will discuss the generalization of both formulations of the scheme to two and higher dimensions and to systems. Again the difference and relative advantages of both formulations will be discussed. For systems, the necessity of using local characteristic decompositions will be illustrated, together with some recent attempts to make this part of the algorithm, which accounts for most of the CPU time, cheaper. Lecture 5: Practical issues Practical issues of the ENO and WENO algorithms, such as implementation for workstations, vector and parallel supercomputers, how to treat various boundary conditions, cur-,'ilinear coordinates, how to use artificial compression to sharpen contact discontinuities, will be discussed. ENO schemes applied to Hamihon-Jacobi equations will also be discussed. Lecture 6: Applications to computational physics Application of ENO and WENO schemes to computational physics, including compressible Eulcr and NavierStokes equations, incompressible flow, and semiconductor device simulations, will be discussed.

154 d)

High Resolution Methods for the Approximate Solution of Nonlinear Conservation Laws and Related Equations (5 lectures in English) Prof. Eitan TADMOR (UCLA and Tel Aviv University.)

Outline: The following issues will be addressed. Conservation laws: Scalar conservation laws, One dimensional systems; Riemann's problem, Godunov, Lax-Friedrichs and Glimm schemes, Multidimensional systems Finite Difference Methods: TVD Schemes, three- and five-points stencil scheme, Upwind vs. central schemes, TVB approximations. Quasimonotone schemes; Time discretizations Godunov-Type Methods and schemes Finite Volume Schemes and error estimates Slreamline Diffusion Finite Element Schemes; The entropy variables Spectral Viscosity and Hyper Viscosity Approximations; Kinetic Approximations Lecture 1: Approximate solutions of nonlinear conservation laws - a general overview During the recent decade there was an enormous amount of activity related to the construction and analysis of modem algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present the successful achievements of this activity we discuss some of the analytical tools which are used in the development of the convergence theories associated with these achievements. In particular, we highlight the issues of compactness, compensated compactness, measure-valued solutions, averaging lemma,...while we motivate our overview of finite-difference approximations (e.g. TVD schemes), finite-volume approximations (e.g. convergence to measurevalued solutions on general grids), finite-element schemes (e.g., the streamline-dissusion method), spectral schemes (e.g. spectral viscosity method), and kinetic schemes (e.g., BGK-like and relaxation schemes). Lecture 2: Approximate solutions of nonlinear conservation law - nonoscinatory central schemes We discuss high-resolution approximations of hyperbolic conservation laws which are based on t centraldifferencing. The building block of such schemes is the use of staggeredgrids. The main advantage is simplicity, since no Riemann problems are involved. In particular, we avoid the time-consuming field-by-field decompositions required by (approximate) Riemann solvers of upwind difference schemes. Typically, staggering suffers from excessive numerical dissipation. Here, excessive dissipation is compensated by using modern, high-resolution, non-oscillatory reconstructions. We prove the non-oscillatory behavior of our central procedure in the scalar framework: For both the second- and thirdorder schemes we provide total-variation bounds, one-sided Lipschitz bounds (which in turn yield precise error estimates), as well as entropy and multidimensional L " stability estimates. Finally, we report on a variety of numerical experiments, including second -and third-order approximations of onedimensional problems (Euler and MHD equations), as well as two-dimensional systems (including compressible and incompressible equations). The numerical experiments demonstrate that these central schemes offer simple, robust, Riemann-sulver-free approximations for the solution of one and two-dimensional systems. At the same time, these central schemes achieve the same quality results as the high-resolution upwind schemes. Lecture 3: Approximate solutions of nonlinear conservation laws - convergence rate estimates Convergence analysis of approximate solutions to nonlinear conservation laws is often accomplished by BV or componsated-compacmess arguments, which lack convergence rate estimates. An L l -error estimate is available for monotone approximations. We present an alternative convergence rate analysis. As a stability condition we assume Lip + -stability in agreement with Oleinik's E-condition. We show that a family of approximate solutions, v ,ps which is Lip + -stable,satisfies it v ep,

(..t) - u ( . , t ) II t.,p -< C II v "p, ( . , 0 ) -

u(..O) ~1up

Consequently, familiar L ~ and new pointwise error estimates are derived. We demonstrate these estimates for viscous and kinetic approximations, finite-difference schemes, spectral methods, coupled systems with relaxation... Lecture 4: Approximate solutions of nonlinear conservation laws - the spectral viscosity method Numerical tests indicate that the convergence of spectral approximations to nonlinear conservation laws may - and in some cases we prove it must --- fail, with or without post-processing the numerical solution. This failure is related to the global nature of spectral methods. Since nonlinear conservation laws exhibit spontaneous shock discontinuities, the spectral approximation ponutes unstable Gibbs oscillations overall the computational domain, and the lack of entropy dissipation prevents convergence in these cases.

155 The Spectral Viscosity (SV) method attempts to stabilizethe spectral approximation by augmenting the latmr with high frequency viscosityregularization,which could bc efficientlyimplemented in the spectral,ratherthan the physical spare. The additionalSV is small enough to retainthe formal spectzalaccuracy of the underlying approximation; yet, the SV is shown to Ix:largeenough to enforce a sufficientamount of entropy dissipation,and hence, by compensated compactness arguments, to prevnm unslable spurious oscillations.Rec.cnt convergence results for the SV approximations of initialand initial-boundaryvalue problnms will be surveyed. Numerical experiments will bc presented to confLrm that by postprocessing the $V solution,one recovers the exact entropy solution within spectralaccuracy. Lecture 5: Approximate solutions of nonlinear conservation laws: entropy, kinetic formulations and regularizing effects We present a new formulation of multidimensional scalar conservation laws and certain 2x2 one-dimensiomd systems (including the isentropic equations), which includes both the equation and the entropy criterion. This formulation is a kinetic one involving an additional variable callecl velocity by analogy. We also give some applications of this formulation to new compactness and regularityresults for en~opy solutions based upon the velocity averaging lemmns. Finally,we show that this kinetic formulation is in fact valid and meaningful for more general classes of equations likeequations involing nonlinear second-order terms.

FONDAZIONE C.I.M.E. CENTRO INTERNAZIONALE MATEMATICO ESTIVO INTERNATIONAL MATHEMATICAL SUMMER CENTER

"Arithmetical Theory of Elliptic Curves" is the subject of the Third 1997 C.I.M.E. Session. The session, sponsored by the Consiglio Nazionale delle Ricerche (C.N.R), the European Community under the "Training and Mobilty of Researchers" Programme and the Ministero dell'Universith e della Ricerca Scientifica e Tecnologica (M.U.tLS.T.), will take place, under the scientific direction of Professor CARLO VIOLA (Universit~ di Pisa) at Grand Hotel San Michele, Cetraro (Cosenza), f r o m 12 t o 19 J u l y , 1997. Courses

a) I w a s a w a T h e o r y for E l l i p t i c C u r v e s W i t h o u t (6 lectures in English). Prof. John COATES (Cambridge University)

Complex Multiplication.

Outline: Let E be an elliptic curve over Q without complex multiplication, and let p be a prime number. A considerable amount is now known about the Iwasawa theory of E over the cyclotomic Z v extension of Q, and some of this material will be discussed in the courses by Greenberg and Rubin. On the other hand, very little is known about the Iwasawa theory of E over the field F~r which is obtained by adjoining all p-power division points on E to Q. If G~o denotes the Galois group of F ~ over Q, then Gor is an open subgroup of GL2 (Zv) by a theorem of Serre, since E is assumed not to have complex multiplication. The course will begin by discussing some of the basic properties of the non-abelian Iwasawa algebra A -- Z v [[Gcr 9 It will then give our present fragmentary knowledge of various basic I ~ s a w a modules over A, which arise when one studies the arithmetic of E over F c r Much of the course will be concerned with posing open questions which seem to merit further study.

Reyerenr.es The course will only assume basic results about elliptic curves and their Galois cohomology, most of which are contained in - J. Si]verman, "The Arithmetic of Elliptic Curves", GTM 106, (1986), Springer.

b) I w a s a w a T h e o r y for E l l i p t i c C u r v e s . (6 lectures in English) Prof. Ralph GREENBERG (University of Washington, Seattle)

Outline: This course will present some of the basic results and conjectures concerning the behavior of the Selmer group of an elliptic curve defined over a number field F in a tower of cyclotomic extensions of F. We will mostly discuss the case where the elliptic c~trve

157

has good ordinary reduction at the primes of F dividing a rational prime p and the cyclotomic extensions are obtained by adjoin p-power roots of unity of F. One of the ma]u results we will prove is Mazur's "Control Theorem". The proof will depend on a rather simple description of the Selmer group which also will allow us to study specific examples. We will also discuss the :'Main Conjecture" which was formulated by Mazur and its relationship to the Birch and Swinnerton-Dyer Conjecture. If time permits, we will discuss generalizations of this Main Conjecture.

ReJerences 1) J. S[Iverman~ The Arithmetic of Elliptic Curves, GTM 106, Springer. 2) S. Lang, Cyclotomic Fields I and II, GTM 121, Springer. 3) L. Washington, Introduction to Cyclotomic Fields, GTM 83, Springer. 4) J. P. Serre, Cohomologie galoisienne, LN.~vI 5, Springer. 5) J. Tare, Duality Theorems in Galois Cohomology over Number Fields~ Proceedings of the ICM, Stockholm, 1962, pp. 288-295. In reference 1, one should become familiar with the Tate modale of an elliptic curve and reduction modulo a prime p. In references 2 or 3 one can find an introduction to the structure of finitely generated modules over the Iwasawa algebra. It may be useful to consult references 4 and 5 to become familiar with Galois cohomolo~-. c) T w o - d i m e n s i o n a l r e p r e s e n t a t i o n s o f G a l ( Q / Q ) . (6 lectures in English) Prof. Kenneth A. RIBET (University of California, Berkeley)

Outline: The general theme of the course is that two-dimensional representations of Gal (-Q/Q) can be sho~n to have large images in appropriate circumstances. The lectures will touch on topics to be selected from the following list: 1. Recent work by Darmon and Mere! on modular curves associated to normalizers of non-split Cartan Subgroups of G L ( 2 , Z / p Z ) . A preprint by these authors proves, among other things, that the Fermat-like equations xp § yp _- 2z p has only the trivial nonzero solutions in integers x, y and z when p is an odd prime number! Technically, the main theorem of the paper is an analogue of a result pertaining to split Cartan subgroups of G L ( 2 , Z / p Z ) which appears in B. Mazur's 1978 paper "Rational i~ogenies o] prime degree". The connection with two-dimensional representations of Gal(-Q/Q) comes about because Merel has used the Darmon-Merel theorem to prove that the rood p representations of G a l ( Q / Q ) arising from non-CM Frey elliptic curves have large images for p > 2. [To read about this work prior to the conference, d o ~ l o a d Darmon-Merel paper (http://w~r~. math. m c g i l l , c e d d a r m o n / p u b / W i n d i n g / p a p e r , html) and follow the references given in that article] .2. Semi.stable ~epresentations. In his 1972 article on Galois properties of torsion points of elliptic curves, J.-P. Serre proved that the nmd p representation of Gal(Q/Q) associated to a semistable elliptic curve over Q is either reducible or surjective, provided that the prime p is at least 7. In his 1995 Bourbaki seminar on the work of Wiles and TaylorWiles, Serre extended his theorem to cover the primes p = 3,5. Using group-theoretic results of L. E. Dickson, one may replace Z / p Z by a finite etale Z / p Z algebra. [See ( f t p : / / m a t h . b e r k e l e y . e d u / p u b / P r e p r i n t s / K e n - R i b e t / s e m i s t a b l e . rex).]. 3. Adelic representations. A recent theorem of R. Coleman, B. Kaskel and K. Ribet concerns point P on the modular curve X0 (37) for which the class of the divisor (P) (Poo) has finite order. (Here Por is the "cusp at infinity" on X0(37).) The theorem states that the only such points are the two cusps on X0 (37). Our proof relies heavily on results of B. Kaskel, who calculated the image of the "adelic" representation Gal(Q/Q) GL (2, R) which is defined by the action of G a l ( Q / Q ) on the torsion points on the Jacobian of Xo (37). Here R is the ring of pairs (a, b) E Z x Z for which a and b have the same

158

parity. In my lectures, I would like to explain how Kaskel arrives at this results. [Although neither Kaskel's work nor the ColemA--Kaskel-Ribert is available at this time, one might look at 6 of Part I of the Lang-Trotter book on "~obenius distributions" (Lecture Notes in Math, 504) to get a feel for this kind of study]. 4. l-adic representation attached to m o d u l a r f o r m s . L e t f l, . . . , f p be newforms of weights k l , . . . , kp possibly of different levels and characters. Let E = E1 x . . . x F_~ be the product of the number fields generated by the coefficients of the different forms. For each prime , the l-adic representations attached to the different fi furnish a product representation p~ : G a l ( - Q / Q ) ~ G L (2, E | Ql). in principle, we know how to calculate the image of p "up to finite groups", i.e., the Lie algebras of pGa/(Q/Q). I hope to explain the answer - and how x~e know that it is correct. [One might consult my article in Lecture Notes in Math. 601 and my article in Math. Annalen 253 (1980)]. d) Elliptic curves w i t h c o m p l e x m u l t i p l i c a t i o n and t h e conjecture of Birch and Swirmerton-Dyer. (6 lecture in English) Prof. Carl RUBLN (Ohio State University) Outline:

The conjecture of Birch and Swinnerton-Dyer relates the arithmetic of an el]iptic curve with the behavior of its L-function. These lectures will give a survey of the state of our "knowledge of this conjecture in the case of elliptic curves with complex multiplication, where the results are strongest. Specific topics will include: 1) Elliptic curves with complex multiplication. 2) Descent (following Coates and Wiles). 3) Elliptic units. 4) Euler systems and ideal class groups. 5) Iwasawa theory and the "main conjecture". ~ e f erence$

- Shimura. G., Introduction to the arithmetic theory of automorphlc functions, Princeton: Princeton Univ. Press (1971), Chapter 5. - Silverman, J., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Math. 151, New York: Springer-Verlag (1994), Chapter II. - de Shaht, E., Iwasawa theory of elliptic curves with complex multiplication, Perspectives in Math. 3, Orlando: Academic Press (1987). - Coates, J., Wiles, A., On the conjecture of Birch and S~innerton-Dyer, Invent. math. 39 (1977) 223-251. - Rubin, K., The main conjecture. Appendix to: Cyclotomic fields I and II, S. Lang, Graduate Texts in Math. 121, New York: Springer-Verlag (1990) 397419. Rubin, K., The 'main conjectures' of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991) 25-68. -

FONDAZIONE CTM.E. CENTRO INTERNAZIONALE M A T E M A T I C O ESTIVO INTERNATIONAL M A T H E M A T I C A L SUMMER CENTER

"Quantum Cohomology" is the subjectof the Fourth 1997 C.I.M.E. Session. The session,sponsored by the Consiglio Nazionale delle Ricercbe (C.N.R), the Ministoro dell'Universit,~t9 della Ricerca Scientifica9 Tecnologica (M.U.R.S.T.), and European Community will take place, under the scientificdirectionof Professors P A O L O D E B A R T O L O M E I S (Universiu3 di Firenzc), BORIS D U B R O V I N (SISSA, Trieste) and C E S A R E R E I N A (SISSA, Trieste)at Grand Hotel San Michele, Cen..aro(Coscnza), from June 30 to July 8,1997.

Courses

a)

Recent Developments in Non Pertm'bative Aspecet of String Theory (8 lectures in English). Prof. C~sar G O M ] ~ (Institutode Matemlticas y FfsicasFundamental, Madrid)

The aim of the course will be to review recent developments in non perturbative aspects of su'ing theory. The main topics we will cover are: i) Heterotic type 11 dual pairs and quantum feld theory limits. We will concentrate our study in Calabi Yau manifolds which are K3 fibrations or elliptic fibradons. We will derive the exact Seiberg Witten solutions from the local geometry of the Ca]aN Yau manifold. ii) Mirror symmetry for K3. Picard lattice, geometric mirror and T duality. We will consider some examples related with D brahe probes. iii)D iestamons associated with divisors of arithmetic genus one, and non perturbative superpotentials in 3 and 4 dimensional susy gauge theories.

References I) S. Kachru and C. Vafa, Nucl. Phys. B450, 1969, 69. 2) P. Aspinwall, K3 surfacesand stringduality,bepth 9611137. 3) E. Witten, Nucl. Phys. B474, 1966, 343.

b)

I. 2. 3) 4) 5) 6)

Gromov-Witten Invariants and Some Appficatlom to Symplectlc Topology, Enumerative Geometry, Finer Theory and Hmniltonian Systems (8 lecture in English) Prof. Gang TIAN 0d.I.T.,Cambridge)

In thisseriesof lectures,the following topicswill be covered: Introductionto symplectic manifolds and pseudo-holomorphic curves; Moduli space of holomorphic stablemaps; Gromov-Witten invariantsof general symplectic manifolds; Associadvity of quantum cohomology; Some applicationsto symplectic topology and enumcrative geomcWy; Relationto Flocr theoryand Harniltoniansystems.

The material should be enough for eight lectures.If extra time remains something about mirror symmetric for Ca]abi-Yau manifolds willbe intzoduced.

References I.

K. Behrend, Gromov-Witten invariantsin algebraic geometry, preptint,alg-georn/960101 I.

160

2. 3. 4.

14. 15.

K. Behrend and B. Fantechi,The inu'insicnormal cone, preprint,alg-geom/9601010. K. Fukaya and K. Ono, Arnold conjecture and Gromov-Wittcn invariants,prcprint, 1996. M, Kontsevich and Y. Martin, GW classes,Quantum cohomology and enumerative geometry, Comm. Math. Phys. 164, (1994), 525-562. J.l.,i and G. Tian, Vh'mal moduli cycle and Gromov-Witteo invariants of algebraic varieties, preprint, alggeom/9602007 J. Li and G. Tian, Virtual moduli cycle and Gromov-Witten invariants of general symplectic manifolds, preprim (alg-geom), August, 1996. D. McDuff and S. Salamon, J-holomorptuc curves and quantum cohomology, University Lec. Series, vol. 6, AMS. Y. Ruan, Topological sigma model and Donaldson-type invariants in Gromov theory, Duke Math. J. 83, (1996), 4t51-500. G. Liu and G. Tian, Arnold conjecture for general symplectic manifolds, prcprint, August, 1996. Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Diff. Geom. 42, no.2, (1995). B. Siebert, Gromov-Witten invariants for general sympleetic mamfolds, preprint, 1996. C. Taubes, The Seiberg-Wittcn and the Gromov invariants, J. of Amer. Math. Sot:., (1996). G. "lima, Quantum eohomology and its associativity, Proc. of Ist Current developments in Mathematics, Cambridge, 1995. F,. Witten, Topological sigma models, Comm. Math. Phys. 118, (1988). E. Wittcn, Two dimensional gravity and intersection theory on modu/i space, Survey in Diff. Geom., 1992.

c)

Elliptic Quantum Groups and Algebraic Bethe Ansatz (8 lectures in English)

5. 6. 7. 8. 9. 10. 11. 12. 13.

Prof. Alexander VARCHENKO (University of North Carolina at Chapell Hill) The Bethe ansatz is a method to construct common eigenvcctors of commuting families of operators (transfer matrices) occurring in two-dimensional models of statisticalmechanics. Faddcev and Takhtadzhan [6] reformulated the problem as a question of r~presentationtheory: commuting families of transfermatrices are associated to representations of era'rain algebras with quadratic relations (now called quantum groups). Eigenvectors arc constructed by properly acting with algebra elemeats on 'q'tighcstweight vectors", In this form, the Bethe ansatz is called the algebraic Bethe Whereas this construction has been very successful in rational and trigonometric intcgrable models, its extension to ellipticmodels has been problematic, although the Bethe ansatz, in the case of the eight-vertexmodel, is known since Baxter's work [I]. Recently, a definitionof ellipticquantum groups Ex.n (O) associated to any symple classicalLie algebra g was given by Fr [2]. It is related to a q-deformation of the Knizlmik-Z, amolodchikov-Bcrnard equation on toil.The representationtheory of E~.~ (sl~) was described in [3] and the algebraic Bethe ansatz for E~,~ (sl~) was intxoducedin [4]. The conslructionis very close to the construction done in the trigonometric case in [6].The main difference is that transfermatrices act on spaces of vector-valued functions rather than on finitedimensional vector spaces. Another class of problems in which the Bethe ansatz has been applied successfully is the class of C.alogeroMoscr-Sutherlaod quantum many body problems on the line with ellipticpotentials.In the case of two bodies, the Bethc ansatz goes actuallyback to Hermitc, who solved in this way the generalized I..am~ equations, cf. [7].These integrable SchrSdinger operators admit a q-deformation due to Rujsenaars [5]. In the lcctmv.s we shall discuss elliptic quantum groups, their connections with two-dimensional models of sutfi.Cdealmechanics, Kaiz.hnik-Zamolodchikov equations, and the Rujsananrs operators.

References [I]

R.J. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropie spin chain I, I1 and IlL Ann. Phys. (N. Y.) 76, 1-24, 25-47, 48-71 (1973). [2] G. F'clder, Conformal field theory and integrable systems associated to elliptic curves, Proceedings of the Intexnational Congress of Mathematicians, Ziirich 1994, p. 1247-1255, Birkh~iuser, 1994; Elliptic quantum groups, preprint hep-th/9412207, to appear in the Proce.edings of the IC'MP, Paris 1994. [31 G. Fr and A. Varehenko, On representations of the elliptic quantum group Et. q (sl l ), CMP, 1996. [4] G. Feldcr and A. Varchenko, Algebraic Bethe ansa~ the elliptic quantum group E t .n (el 2 ), Nuclear Physics B 480 (1996), 485-503. [5] $. ~. M. Ruijsenaars' C~mplete integrabi~ity ~f relativistic Calager~-M~ser ~.stems and ~1lipticfuncti~n /dem/t/es, Commun. Math. Phys. 110, 191-213 (1990). [6] L.A. Takhtadzhan and L. D. Faddccv, The quantum method of the inverse problem and the Heisenberg XF'Z chain, Russ. Math. Surv. 34/5, 11-68 (1979). [7] E.T. Whittaker and G. N. Watson, A course o f modern analysis, Cambridge University press, 1915.

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