Proceedings of the International Conference on Modern Mathematics and the International Symposium on Differential Geometry in Honour of
0 W s f i w y < hi/ £BucAif7/ on the Centenary of His Birth
DIFFERENTIAL GEOMETRY HID RELATED TOPICS
Editors
Gu Chaohao Hu Hesheng Li Tatsien
World Scientific
DIFFERENTIAL GEOMETRY HMD » M H TOPICS
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Gu Chaohao Hu Hesheng L>i liltSIGri Fudan University, China
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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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Preface The International Conference on Modern Mathematics and the International Symposium on Differential Geometry in honor of Professor Su Buchin for the centenary of his birth was held from September 19 to September 23, 2001 at Pudan University, Shanghai, China. Around one hundred mathematicians from China, France, Japan, Singapore and the United States participated in the conference and the symposium. We wish to express our sincere thanks to all the invited speakers and the participants for their participation and contribution. The academic program of the International Conference on Modern Mathematics includes 14 plenary lectures and 15 lectures in specialized parallel sessions. For the International Symposium on Differential Geometry, there are 8 plenary lectures and 12 lectures in parallel sessions. This Proceedings collects 20 comprehensive lectures and original papers. The contents cover a broad spectrum of advanced topics in mathematics, especially in differential geometry, for example, some problems with common interest in harmonic maps, submanifolds, Yang-Mills field and the geometric theory of solitons etc. The participants of the symposium extended their highest respect to Professor Su Buchin for his outstanding contributions in mathematics, and wish him good health and a long life. Both the conference and the symposium were sponsored by - China Society of Industrial and Applied Mathematics - Chinese Academy of Sciences - Chinese Association for Science and Technology -
Chinese Mathematical Society Fudan University Mathematical Center of Ministry of Education of China Ministry of Education of China National Natural Science Foundation of China
- Shanghai Association for Science and Technology - Shanghai International Civilization Association.
Gu Chaohao v
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Contents
Preface
v
The Mobius equivalent isothermic surfaces in S3(l) Backlund transformations J. H. Chen and W. H. Chen
and
Asymptotic behavior of Yang-Mills flow in higher dimensions Y. M. Chen, C. L. Shen and Q. Zhou The essential spectrum on complete noncompact Riemannian manifolds with asymptotically nonnegative radial Ricci curvature Z. H. Chen and C. H. Zhuo Complete submanifolds in Euclidean spaces with constant scalar curvature Q. M. Cheng On mathematical ship lofting G. C. Dong, A. X. Hong and Z. C. Liu
1
16
39
48
64
Some developments in spectral methods for nonlinear partial differential equations in unbounded domains B. Y. Guo
68
The Darboux transformation and local isometric immersions of space forms Q. He and Y. B. Shen
91
On the Nirenberg problem M. Ji
107
Periodic mean curvature and Bezier curves K. Kenmotsu
135
VII
VIII
Almost complex manifolds and a differential geometric criterion for hyperbolicity S. Kobayashi
147
Thom isomorphism of equivariant cohomology X. M. Mei
157
Horizontally conformal F-harmonic maps X. H. Mo and C. H. Yang
166
Harmonic maps between Carnot spaces S. Nishikawa
174
Harmonic cohomology groups on compact symplectic nilmanifolds Y. Sakane and T. Yamada
204
A survey of complete manifolds with bounded radial curvature function K. Shiohama
216
Yang-Mills connections over compact symplectic manifolds H. Urakawa Nonlinear Schrodinger systems associated with Hermitian symmetric Lie algebras H. Y. Wang On the Hensel lift of a polynomial Z. X. Wan Some advances related particle systems — Estimation for eigenvalue and new ^ | n e l d S. J. Yan A note on locally real hyperbolic space with finite volume Y. H. Yang
227
237
250
257
264
ix
Interesting properties of the sets: A ^ l 2 , 2 2 , 3 2 , . . . ] , JV3[13, 2 3 , 3 3 ,...] and JV4[14, 2 4 , 3 4 ,...] Y. Q. Ye
272
List of participants
279
T H E M O B I U S EQUIVALENT ISOTHERMIC SURFACES I N S 3 ( l ) AND BACKLUND TRANSFORMATIONS
JIANHUA CHEN School of Mathematics, Peking University, Beijing 100871, China Armored Force Engineering Institute, Beijing 100072, China WEIHUAN CHEN School of Mathematics, Peking University, Beijing E-mail:
[email protected] 100871,
China
In this paper we study the surfaces in S 3 ( l ) in the sense of Mobius geometry following C.P.Wang's paper [14]. We classify the isothermic surfaces with constant Mobius Gaussian curvature. On the other hand we find out a class of special isothermic surfaces which allow Backlund transformations for their integrable equations.
§0 Introduction The modern theory of completely integrable systems can be traced back over a century to the classical differential geometry. For instance, one of the most important ingredient of soliton theory is the theory of Backlund transformation. And also the modern approach of integrable system theory promopts the development of differential geometry (e.g. [1], [2], [3], [6], [9], [10], [11]). In this paper, we try to study the integrable systems of the surfaces in S 3 (l) in the sense of Mobius geometry. A completely invariant system for a surface in S"(l) under the action of Mobius group is given by C.P.Wang in [14]. By this invariant system, we study the integrable conditions of isothermic surfaces in 5'3(1). We realize that the theory of the isothermic surface in 5 3 (1) can be reformulated within the modern approach of soliton surface (see [6]). The isothermic surface in 5 3 (1) is determined totally by a PDE of the Ath order. It is hard to obtain its solution. However the surfaces with special Mobius Gaussian curvature are determined totally by a PDE of the 2nd order. We give their Backlund transformations via the Riccati forms of the Lax pairs.
1
2
§1 The integrable conditions for the surfaces in S 3 ( l ) Let i?f be the Lorentz space, i.e. R5 with the inner product (, ) given by (X, Y) := -x02/o + xiyi + x2yi + x3y3 + x 4 y 4 , where X - (xQ,xi,x2,x3,Xi), Y = (2/0,2/1,2/2,2/3,2/4) £ # 5 - We denote by C\ the half cone in R\ and by Q 3 the quadric in RPA: C\ := {X G Rl\(X,X)
= 0,x0 > 0},
Q3:={[X]£RP4\(X,X)=0}. Let 0(1,4) be the Lorentz group of R\, which keeps the inner product (, ) invariant, therefore keeps C\ invariant. Then (9(1,4) induces a transformation group on Q3 whose action is defined by T([X]):=[XT\,
XeCX,
Te
0(1,4).
Let x(P)(P G M) be a point in S 3 (l) c R4. Then x(P) • x(P) = 1, where the left-hand side means the Euclidean inner product of x(P) with itself. Put X = (l,x(P)), then <X>A") = - l + a ; ( P ) - a : ( P ) = 0 , so X £ C+. Let 7r: C^. -> Q 3 be the natural projection, then S 3 (l) can be identified with Q 3 . It is evident that, under the above identification, the trasformation of T G 0(1,4) acting on Q3 is a Mobius transformation on 5 3 (1) which keeps hyperspheres invariant. Conversely, each Mobius transformation on 5 3 (1) can be represented by a transformation of some T G 0(1,4) on Q 3 . Therefore two surfaces x, x : M —> 5 3 (1) are Mobius equivalent if and only if there exists a T G 0(1,4) such that [l,x] = T([l,x]) : M -> Q3 (see [14]). Let x : M -*• S 3 (l) be a surface in 5 3 (1) and y : U -> C\_ be a local lift of the map [l,a;] : M —t Q3 defined on an open set U C M. Since y = p(i,x) for some positive function p defined on U, we know that (dy, dy) = p2dx-dx is an induced Riemannian metric on U from R\. Denote by A and by K the Laplacian operator and its Gaussian curvature with respect to this metric. Then g=((Ay,Ay)-4K)(dy,dy)
(1.1)
is independent of the choice of local lifts. (For details, cf. [14]). Thus it is globally defined on M. It is clear that g is Mobius invariant under Mobius transformation and thus it is called the Mobius metric on M. By a calculation the Mobius metric on M can be expressed as 9 = 2\\II - Hlfdx
• dx,
3
in which I and II are the first and the second fundamental forms of the surface x : M -* 5 3 (1) respectively, and H is its mean curvature. Therefore the Mobius metric is nondegenerate around any non-umbilical point. We assume that the surface x : M -> 5 3 (1) has no umbilic point. Then (Ay, Ay) - 4K > 0 there exists a unique lift Y : M ->• C% such that = \2dx-dx.
g = (dY,dY)
(1.2)
3
which is called the canonical lift of x : M -> 5 (1), and (1.2) means that the lift Y : M —> C+ C R\ is an isometric immersion of M with the Mobius metric g into R\. Let {a;;.Ei,£2} be a local orthonormal frame field on M with respect to g and UJI,CJ2 be its dual coframe field. Define N := ~ A Y - i ( A F , AY)Y,
Y, := ^ ( Y ) = Y , ^ )
(j = 1,2),
where A is the Laplacian with respect to g. Then (Y,Y)=0,
(Y,Yj)=0,
(Yi,YJ)=6ij>
(N,Y) = 1, (N,N) = (N,Yj) = 0, Thus at each point of M, V = {Span{Y,N, positive definite subspace in R^ such that
jf = 1,2.
Yi,Y2}}J-
Rl = Span{AT, Y} © Spa,n{YuY2}
is a 1-dimensional
© V.
Let E be a section to V with ( £ , E) = 1. Then {x; Y, AT, Yx, Y2, E} forms a moving frame in flf along M, in which Yi are tangent to the lift Y : M -» C+ and Y, A'', £ are normal vectors. The Weingarden-type formulae of (M,g) are given as follows (cf. [14]): N
0 0
0 0
E ^u i
Yi
-E^ij^i
-Wi
0
Y2 \Ej
-E^2jWj
—Cd2
- E c>,-
0
Wl
U>2
w
-W12
0
E ^2jWj £ Cj"j W12 E -Sij^j 0 T,B2jUj
- £ BijWj -T.B2iu,i
0
AT Yi Y2
/
VW
(1.3) where B y ( = Bji) and Cj are respectively the components of the Mobius second fundamental form and the Mobius form of a;, Aij are the components of another symmetric tensor on M. Furthermore Bij = A (hij — HSij),
£^H>
£*« = °-
where (hij) is the second fundamental form of x : M s
k E ha i its mean curvature.
3
(i-4)
S (l) and H =
4
If we take orthogonal net of curvature lines as coordinates (u, v) and choose local orthonormal frame field {x\ E1.E2) such that E1.E2 are prin1 cipal directions on M, then B12 = 0. Prom (1.4) we get Bn — ^ and B22 = — h • Also we have wi = adu and u2 = bdv for some functions a and b. Then (1.3) become N Y1 Y2
fY\ N Yi Y2
= (Pdu + Qdv)
\EJ
(1.5)
\EJ
where
/ P =
/
0 a 0 0 \ 0 Aua A12C1 Cia 0 -Ana-a 0 - ^ \a -A12a 0 %0 0
°
\ -Cia
0 -|o
0
Q =
0 0
0 0 b 0 \ 0 ^126 A22& C2&
-A126 0
0
-A22b-b-%
0 /
\ -C2b
0
0
a 0 1, 26
0
1, °
/
The integrable conditions of (1.5) is Pv-Qu
+ [P,Q]=0.
(1.6)
which is equivalent to the following system of equations = - l j „ , C 2 = i o . , A„ = - i ( m ( « l - ) ) m ,
(1.7)
(y).+ (£). =-**— Bs[ is an isometric map of the surface with Mobius metric into i?f, from (1.3) we have dY = wiYi + u2Y2, and j = (dy, d y ) = w? + w^. Because Y, N, E are normal to the surface, we also have DYt = (dY1)T
= CJ12Y2,
DY2 = (dY2)T = - w i 2 y i . So W12 is the form of Levi-Civita connection on M with respect to the Mobius metric g. §2 The isothermic surfaces in S 3 ( l ) Recently there are many interesting results about the isothermic surfaces [2, 5, 6, 9]. We will come to discuss the isothermic surfaces in 5 3 (1) by the theory of integrable system. So called an isothermic surface x : M —> S3(l) means that there exist local coordinates u, v around each point on M such that u, v are isothermic coordinates and they also give the orthogonal net of curvature lines on M. We call such a kind of parameters (u, v) as isothermic coordinates on an isothermic surface M in 5 3 (1). It is evident that isothermic coordinates (u, v) on a isothermic surface without umbilics are determined up to a translation and a homothetic transformation, i.e., (u,v) are also isothermic coordinates if and only if u = cu + a, v = cv + b, where a, b and c are constant and c^O. Let x : M —> SZ{1) be an isothermic surface. From (1.2) and (1.4), we know that there exist isothermic coordinates (u,v) such that the Mobius metric g is given by g = e2"{du2+dv2),
(2.1)
and B\i = -B22 = \,
B12=0.
(2.2)
Let Ei1 = e " w # and E2 = e^-S-. Then wi = e"du and w2 = eudv. The ou ov Levi-Civita connection (Ji2 for g is 0J12 = —cjvdu + ujudv.
(2.3)
6
From (1.8) we know uvv + uiuu = -Ke2u.
(2.4)
Then (1.11) and (1.12) become (Fv= -Kve2w - Auuuv - uive2w 2u \FU= Kue2" + 4uvvu + uue . The integrable condition of (2.5) is Kuv + Kuu>v + KVUJU + 2e~2u(u)Uuuv + uvvvu) + u>uv + 2u>uuv = 0. (2.6) Prom (2.4) we get -e_2w(UJUUUV +Uvwu) = Kuv + 2Kuuv + 2Kvuu + 2K(uuv + 2UJUUJV). (2.7) Therefore (2.6) becomes Kuv + 3LJVKU + 3UJUKV + (4K - l)(w„„ + 2uuuv) = 0.
(2.8)
Furthermore (2.4) and (2.8) yield +w„(w m t l 4- uJuw) + e2u(u}uv + 2LJUOJV) = 0, which is the integrable condition of (1.5) for the isothermic surfaces in S 3 (l). In other words, every solution of (2.7) is correspondence to an isothermic immersion x : M —> 5 3 (1) (up to a transformation in 0(4,1)). Exactly, we have the following theorem. T h e o r e m 2.1 If x : M ->• 5 3 (1) is an isothermic immersion, and its Mobius metric is g = e2ljj{du2 + dv2), where (u,v) are isothermic coordinates, then u is a solution of (2.9). Conversly, if ui is a solution of (2.9), then there exists an isothermic immersion x : M —• S 3 (l) such that (u,v) are its isothermic coordinates and g = e2w(du2 +dv2) is its Mobius metric. Proof. We only need to prove the second part of theorem. Let LO be the solution of (2.9) and put K := — e~2u(ojuu + uvv). Since (2.9) is the integrable condition of (2.5), we get F — F(u,v) by solving the system of (2.5). From that An - A22 = e"2wF and An + A22 = ^ + K, we get An and A22. Set C\ = -wue~" and C2 = wt,e~"J,^12 = 2uuve~2u. Then equations (1.7) ~ (1.10) hold for a = b = ew, i.e., the equations (1.3) are completely integrable for Bn = —B22 = A, £12 = 0. Let (Y,N,Yi,Y2,E) be a solution of (1.3), in which Y,N,YUY2,E are smooth mappings from a domain U of (u,v) into iZf. It is evident by a standard argument that the solution (Y, N, Yi, Y2, E) satisfies conditions 0 = (Y,Y) = (N,N) = 0,(N,Y) = 1, and (Y,Yj) = (N,Yj) = (Y,E) = (Yj,E) = (N,E) = 0,(Yi,Yj) = 6ij,Y e C\, if they are true for its initial value (Y0,N0, {Yi)0, (Y2)0,E0) at a point (u 0 ,^o)-
7
Denote Y = A(l,a;), then x : U —> 5 3 (1) is an immersion with Mobius metric g = (dY,dY) = e2"{du2 + dv2) To show (x, g) is isothermic, it is enough to show N* := — A AY — A (AY, AY)Y = N, where A is the Laplacian with respect to g. In fact, the Mobius Gaussian curvature of (x, g) is just K and (AF, AY) = 1 + 4/f. (cf. [14]). Hence A y = -[K + \)Y - 27V*. On the other hand, from (1.3), we get YYijUj := dYi - HwijYj = -HAijWjY
- HSijUijN + UBijUjE.
Furthermore A y = -(An
+ A22)Y -2N
= -{K + hjY - 2N.
This shows N* = N. In the case that the Mobius Gaussian curvature is only the function of w, the integrable conditions of (2.9) is of a special form. Lemma Let x : M —> 5 3 (1) be an isothermic immersion, and its Mobius metric be g = e2iJ(du2 +dv2), where (u,v) are isothermic coordinates. If the Mobius Gaussian curvature K of x is a function of ui only, then the equation (2.9) is equivalent to UK' + 4K-l)e2^u=p(u), 2 [Z W> \(K' + 4K-l)e »ujv=q(v), for some functions p(u) and q(v). Proof. Differentiating (2.4) in u and v, we get f uuuu + Uwu = -(K'(UJ) + 2K(aj))e2ucju, (2-H) \uuuv +LJVVV = -(K'(CJ) +2K{u))e2uu)v. Thus for uiu ^ 0 and w„ ^ 0, we have U)u{bJ uuv •+" ^vvv)
—
w
n ( w u « t i "+"
^vvuj-
Obviousely, it is true for the case that wu = 0 or w„ — 0. Then the equation (2.9) turns out to be
{
(Wuu + Wvv)uv + 2(Du(uiuu + CJVV))V + 7y(e W)uv
=
0>
(bJuu + wvv)uv + 2(uiv(u>uu + iovv))u + ^(e "Ow = 0. Hence, there exist the functions p(u) and q(v) such that f (Uuu + uvv)u + 2uu(u)uu + uvv) + \(e2u)u - -p(u), \ (wu« + t*>vv)v + 2uv{uuu + u)vv) + \{e2u)v = -q(v). Putting (2.4) and (2.11) into (2.12), we obtain (2.10). Now we consider two special cases from which some interesting results can be drown out.
Theorem 2.2 Suppose x : M —> 5 3 (1) is an isothermic surface, and its Mobius Gaussian curvatue K = const., then its Mobius meric must be one of the followings, in which (u, v) are isothermic coordinates: 1. ds2 = du2 + dv2,
K = 0;
2. ds2 = eu(du2+dv2),
K = 0;
3. ds2 = „ 1 l 2 (du2 + dv2), K cosh u
4. ds2 = -jK(du2 5. ds2 = —~^-{du2 6- ds2 = --~n~{du2
K>0;
+ dv2),
K < 0;
+ dv2),
K < 0;
+ dv2),
K < 0;
7. ds 2 = (u2 + v2){du2 + dv2), 8. ds2 = (coshu + cosv)(du2 + dv2),
K = 0; K — 0;
9. ds2 = (/(«) + 5(«))(rfu2 + dv2),
K^O;
where f'(u) ^ 0,g'(v) ^ 0 are given by f / ' » + 4 K / 3 ( u ) - A/ 2 (u) + 2o/(«) + 6 = 0 , \ g'2(v) + AKg3(v) + Xg2(v) + 2ag(v) - 6 = 0 , in which, X, a and b are constant. Proof : For K = const. ^ 5, it follows from (2.10) that e2" = f{u) + g(v), where f'(u) = JJ^p(u),g'(v) = j^q(v), and f(u)+g(v) > 0. The functions f(u) and g(v) shoud satisfy (2.4), i.e., (f(u)+9(v))(f"{u)+g"(v))-f'2(u)-g'2(v)
= -2K(f(u)+g(v))3.
(2.13)
We solve this equation in several cases: (1) Assume f(u) = const, and g(v) = const., then we can take isothermic coordinates (u,v) such that the Mobius metric g = du2 + dv2. In this case, the Mobius Gaussian curvature K = 0. (2) Suppose g(v) = c = const and /(u) ^ const.. Let /i(u) = / ( u ) + ff(u) > 0. Then (2.13) becomes hh"(u)-h'2
+ 2X/i 3 = 0.
Let 2 = ^ M . (2.14) can be written as 2 ^ | + 2i4T = 0. Hence -
=Cl-4Kh.
(2.14)
9
We consider this equation in the following three subcases: (2a) K — 0. Let C\ = m2,m > 0. We get h = citmu. So we can take isothermic coordinates (u, v) such that the Mobius metric g = eu(du2+dv2), and the Mobius Gaussian curvature K — 0. (2b) K > 0. In this case, c\ must be positive. Let c\ = m 2 , m > 0, then h! = ±hy/m2 — 2Kh. Solving this equation, we get h =
m2 K i
C2±mu e C2±mu +
e
'
In this case, we can take isothermic coordinates such that the Mobius metric g = ——^—n— , with Mobius Gaussian curvature k = const., and K ^ -x. 4 if cosh u (2c) (K < 0). In this case, C\ can be positive, zero or negtive. By a similar calculation, we claim that there exist isothermic coordinates (u, v) on the surface such that its Mobius metric can be written as 1 (du2+dv2) K sinh2 u or 1 (du2 + dv2), Ku2.2 or 1 9 = ~ „__ 2 (du2 + dy2) tfcos* with constant Mobius Gaussian curvature K < 0. (3) Suppose f(u) ^ consi. and g(u) ^ const.. Differentiating (2.13) with respect to u and w respectively, we have f'(u)(g"(v)
- / » ) + (/(«) + 5(t>))/"'(u)
= - 6 t f ( / ( u ) + 9(«)) 2 /'(«)
< ? » ( - < ? » + /"(u)) + (/(u) +
