3 Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Tokyo
RE. Burstall
D. Ferus
K. Leschke R Pedit
Conformal
U. Pinkall
Geometry
of Surfaces in S4 and Quaternions
4
1011. Springer I.,
Authors Francis E. Burstall
Franz Pedit
Dept. of Mathematical Sciences University of Bath
Dept.
Claverton Down
University of Massachusetts
of Mathematics
and Statistics
Bath BA2 7AY, U.K.
1542, Lederle
E-mail. fie. burs tall@maths. bath. ac. uk
Amherst, MA 01003, U.S.A. E-mail: franz@gang. umass. edu
Dirk Ferus Katrin Leschke
Technical
Ulrich Pinkall
University of Berlin
Technical
University of Berlin
MA 8-3
MA 8-3
Strasse des 17.
Strasse des 17.
10623
10623
Juni 136 Berlin, Germany
E-mail. ferus@math. tu-berlin.de
Juni 136 Berlin, Germany
E-mail.
[email protected] E-mail:
[email protected] Cover
figure from D. Ferus, R Pedit:Sl-equivariant Minimal Tori in S'
in S3. Math. Z. 204,269-282
and
Sl-equivariant Willmore Tori
(199o)
CatalogIng-in-Publication Data applied for. Die Deutsche Bibliothek
-
CIP-Einheitsaufnahme
Conformal geometry of surfaces in S4 and quaternions / E E. Burstall Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris ; ....
Tokyo
Springer, 2002 (Lecture notes in mathematics; 1772) ISBN 3-540-43Oo8-3
Mathematics
Subject Classification (2000): 53C42,
53A30
ISSN 0075-8434 ISBN 3-540-43008-3
Springer-Verlag Berlin Heidelberg New York
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Preface
This is the first
comprehensive introduction to the authors' recent attempts understanding, of the global concepts behind spinor representations of surfaces in 3-space. The important new aspect is a quaternionicvalued function theory, whose "meromorphic functions" are conformal maps into ff- which extends the classical complex function theory on Riemann surfaces. The first results along these lines were presented at the ICM 98 in Berlin [10), and a detailed exposition will appear in [4]. Basic constructions of complex Riemann surface theory, such as holomoiphic line bundles, holomorphic curves in projective space, Kodaira embedding, and Riemann-Roch, carry over to the quaternionic setting. Additionally, an important new invariant of the quaternionic holomorphic theory is the Willmore energy. For quaternionic holomorphic curves in HP' this energy is the classical Willmore toward
a
better
energy of conformal surfaces. The present lecture note is based
given by Dirk Ferus at the September, 1999, [3). It centers on Willmore surfaces in the conformal 4-sphere HPI. The first three sections introduce linear algebra over the quaternions and the quaternionic projective line as a model for the conformal 4-sphere. Conformal surfaces f : M -+ HPI are identified with the pull-back of the tautological bundle. They are treated as quaternionic line subbundles of the trivial bundle M x H. A central object, explained in section 5, is the mean curvature sphere (or conformal Gauss map) of such a surface, which is a complex structure on M x IV. It leads to the definition of the Willmore energy, the critical points of which are called Willmore surfaces. In section 7 we identify the new notions of our quaternionic theory with notions in classical submanifold theory. The rest of the paper is devoted to applications: We classify super-conformal immersions as twistor projections from (Cp3 in the sense of Penrose, we construct Bdcklund transformations for Willmore surfaces in HPI, we set Up a duality between Willmore surfaces in S3 and certain minimal surfaces in hyperbolic 3-space, and we give a new proof of the classification of Willmore 2-spheres in the 4-sphere, see Ejiri [2], Musso [9] and Montiel [8]. Finally we explain a close similarity between the theory of constant mean curvature spheres in Summer School
on
Differential
on a course
Geometry
R3 and that of Willmore surfaces in
at Coimbra in
EEP1,
and
use
it to construct Darboux
transforms for the latter.
Bath/Berlin, August
2001
Francis
Burstall,
Dirk
Ferus, Katrin Leschke, Pedit, Ulrich Pinkall
Franz
'Table of Contents
I
Quaternions The Quaternions 1.1 The Group S3 .......................................... 1.2
3
Linear
5
..............................................
.......................................
2
2.1
2.2 3
4
.......................................
7
.........................................
9
..................
9
.........................................
11
Metrics
Moebius Transformations
3.4
Two-Spheres
HPI
Vector Bundles
4.3
7
5
3.3
on
HP .
........................
13
......................................
13
...........................................
15
in
S4
on
Quaternionic Vector Bundles Complex Quaternionic Bundles Holomorphic Quaternionic Bundles
The Mean Curvature Sphere 5.1
S-Theory
5.2
The Mean Curvature
5.3
Hopf Fields
5.4
The Conformal Gauss
.............................
15
...........................
18
.......................
20
.............................
23
...............................................
Sphere
23
.............................
24
............................................
27
Willmore Surfaces
Map
..............................
29
........................................
31
6.1
The
6.2
The Willmore Functional
Energy
1
.........
3.2
4.2
6
..............
Projective Spaces 3.1 Projective Spaces and Affine Coordinates
4.1
5
Algebra over the Quaternions Maps, Complex Quaternionic Vector Spaces Conformal Maps
Linear
1
Functional
..................................
................................
Metric and Affine Conformal
Geometry
31
35
..................
39
..............................
39
7.1
Surfaces in Euclidean
7.2
The Mean Curvature
7.3
The Willmore Condition in Affine Coordinates
Space Sphere
in Affine Coordinates
..........
42
.............
44
Table of Contents
VIII 8
Twistor 81 8.2
9
10
Project"ions
......................................
Twistor
Projections Super-Conformal Immersions
.....................................
47
.............................
50
Bhcklund Transforms of Willmore Surfaces 9.1
Bdcklund Transforms
9.2
Two-Step Bdcklund
Willmore Surfaces in S3
..........................
53 57
..................................
61 61
10.2
63
Hyperbolic
2-Planes
....................................
S3 and Minimal Surfaces in Hyperbolic
...............................................
Spherical Willmore Surfaces in HPI 11.1 Complex Line Bundles: Degree and Holomorphicity 11.2 Spherical Will1nore Surfaces Darboux tranforms 12.1 Riccati
equations
12.2 Constant
mean
Appendix 13.1 The bundle L 13.2 Holomorphicity ......
67
.........
67
.............................
71
.......................................
73
.......................................
curvature surfaces in
R3
...................
..................
73 74
79
...........................................
83
..........................................
83
and the
Ejiri/Montiel
theorem
.............
84
.................
87
..........................................................
89
References Index
64
......................
12.3 Darboux transforms of Willmore surfaces 13
53
10.1 Surfaces in S ...........................................
4-Space
12
...............
...................................
Transforms
10.3 Willmore Surfaces in
11
47
...................................
Quaternions
I
Quaternions
The
1.1
symbols i, j,
H
quaternions
The Hamiltonian
k with
i2 ii
non-zero
-ii
--
multiplication
The
=
ik
k,
has
division
a
=
multiplicative algebra over
k2
-kj
=
ao +
=
generated
by
the
_1' ki
i)
ali
a2i
showed in 1877 that
Frobenius
the reals.
+
and each commutative, and a a skew-field,
not
R-algebras
finite-dimensional For the
-ik
=
We have
inverse:
a
=
=
obviously
but
R, C and H are in fact the only and have no zero-divisors. ciative
we
j2
=
associative
is
element
4-dimensional
R-algebra
the unitary
are
the relations
that
are
asso-
element
a3k,
+
al
(1.1)
R,
C-
define a:=
ao
ali
-
a2i
-
-
a3k,
Rea:= ao, Ima:= Note
that,
and that
with
in contrast
the
ali
+
a2i
+
a3k.
complex numbers,
Im
a
is not
Wb_
=
the real vector space H in the obvious identify with R3: of purely imaginary quaternions subspace
W
=
The reals
are
identified
canonical.
imaginary
unit,
number,
with
RI.
embedding equally i,-j,k
The
The
and in
fact
F. E. Burstall et al.: LNM 1772, pp. 1 - 4, 2002 © Springer-Verlag Berlin Heidelberg 2002
way with
R,
and
IMH.
quaternions imaginary any.purely would do the job. Rom now on, however, we shall C C ffff generated by 1, i. C is less
real
b a.
Weshall the
a
obeys
conjugation
complex numbers for the complex qualify of square -1 quaternion usually use the subfield
of the
Quaternions
1
2
Occasionally be written
we
shall
Euclidean
need the
product
inner
on
R4 which
can
as
< a, b >R=
Re(ab)
=
Re(ab)
2
(ab
+
ba).
Wedefine a
>R
a
a,
=
vfa-
d.
Then
jabj A closer
study of
the
=
jal Ibl.
(1.2)
multiplication
quaternionic
displays
nice
geometric
as-
pects. Wefirst usual
finds
one
mention
that
and scalar
vector
for
the quaternion
products
a, b E Im Eff
=
on
multiplication using
a
consequence
R-
we
ba
=
In
(1-3)
have
if and only if Im a and Im b the reals are the only particular,
ab
(1.1)
state
For a, b G H
Lemma1. 1.
we
both the
the representation
R'
ab=axbAs
incorporates
V. In fact,
are
linearly
dependent that
quaternions
the reals.
over
commute with
all
others. 2.
a' a
-1
=
if and only if Jal two-sphere
=
1 and
S2
Proof.
=
Im
a.
Note that
the set
of
all
such
Write
a
=
a',
b
=
aobo
+
=
aobo
+
ao +
C
V
=
+
Y,
aob'
+
a'bo
+
aob'
+
a'bo
+ a'
for
the
=
bo
IMH.
where the prime
denotes
the
imaginary
Then
part.
ab
All
a
usual
the
is
these
follows. vanishes obtain
products, Romthe if
(2).
and
except same
only
if
a
formula is real
x
Y-
R
=
The
1.2
quaternions
The set of unit
S3 in H =
3-sphere
the
i.e.
it
interpret
RI,
Ip
:=
forms
the group of linear
as
E
HI IM12
maps
51b.
< a, b >:=
group is called We now consider
By (1.2)
this
action
product. complement R'
It
scalar
=
the
action
the
obviously
'53
norm on
a
of
compute the differential
H=
map, in fact
SO(3),
-+
y -+
y
R' and, hence, the Euclidean its orthogonal and, therefore, a representation,
P-1 II.H-
...
For p E
7r.
by
pap-1.
1-4
R C H
stabilizes
Im H. We get :
(/-t, a)
H,
H __
X
preserves
,X
us
group Sp(l). of S3 on H given
symplectic
the
S3
Let
We can also the hermitian
px of H preserving
-+
x
multiplication.
under
group
a
1}
=
product
inner
This
3
S'
The Group
1.2
Group S3
S3 and
E
v
Tt'S3
we
get
dl -,ir(v)(a)
=
-
Now y-1v
vaIL-1
commutes with
pap-1vp-1
-
all
a
E ImH
=
if
p(p-'va only if
and
ap-'v)/-t-1.
-
v
=
rp
for
some
real
of S3 diffeomorphism 0, p. manifold orthogonal preserving onto the 3-dimensional SO(3) of orientation this is of R3. Since S3 is compact and SO(3) is connected, transformations all a E ImH if and only if p E R, i.e. if And since pap-I a for a covering. 2:1. It is obvious that antipodal points of and only if M 1, this covering'is and therefore we S3 are mapped onto the same orthogonal transformation, r.
But then
v
=
because
Hence
I
v
ir
is
a
local
=
=
see
that
Rp3
SO(3)- S31111 Wehave of
SO(3).
now
displayed
This
group
the group of unit the is also called
S3 If
we
identify
H = C G)
=
C?,
SP(j)
quaternions spin group: =
we can
S3
-
as
the universal
covering
Spin(3). add yet
SU(2).
another
isomorphism:
4
In
j (a has
I
fact,
Quaternions let
p
=
(a iP) the following +
=
Ito -
+
1-iij
i,8)j.
C-
Therefore
matrix
po, pi
the
representation
AA1 A,.j Because of yopo + /-tlpj.
S' with
yo +
=
+
=-/-tl =
1,
we
(
jLjj
E C.
C-linear with
map
poj =1(-pi)
have
Po tL1
f4l)
Po
E
SU(2).
+
+
AIL
:
a,
C2
to the
respect
Imo
=
Then for
ifil
jpo.
-
C- R we
-4
basis
C2
,
x
have
1-4
fix
1, j of 0:
Algebra
2 Linear
Maps, Complex Quaternionic
Linear
2.1 Since
we
consider for
options
two
spaces to from the
Quaternions
the
over
be
vector
right
spaces V
vector
the skew-field
over
multiplication
the
i.e.
Spaces
quaternions,
there
We choose quaternion
by scalars.
spaces,
of
Vector
vectors
are
multiplied
are
vector
by quaternions
right: V
x
H -+
V, (v, A)
F-+
vA.
dimension, subspace, and linear map work as in the usual algebra. The same is true for the matrix representation of linear maps in finite dimensions. However, there is no reasonable definition The linear for the elementary symmetric functions like trace and determinant: (i) when using 1 as basis for H, but map A : H -+ IH x -+ ix, has matrix matrix (-i) when using the basis j. If A E End(V) is an endomorphism, v E V, and \ E H such that The notions
commutative
of basis, linear
Av then
for
any p E
H\ f 01 A(vp)
If
A is real
then
we
=
vA,
find
=
(Av)p
=
vAp
=
(vp)(p-lAp).
the
eigenspace is a quaternionic subspace. Otherwise it is vector subspace, and we obtain a whole quaternionic 2-sphere of "associated eigenvalues" (see Section 1.2). This is related to the fact that multiplication from the right) is not by a quaternion (necessarily of the In of V. between Iff-linear an H-linear endomorphism fact, space maps vector vector quaternionic spaces is not a quaternionic space itse f. vector Any quaternionic space V is of course a complex vector space, but this structure on depends choosing an imaginary unit, as mentioned in instead 1.1. We shall section have an additional Complex (quite regularly) from the left, and hence commuting with the quaterstructure on V, acting In other words, we consider nionic J E End(V) such that structure. a fixed a
J'
real
=
-
-1.
but
not
a
-
Then
(X
+
iy)v
:=
F. E. Burstall et al.: LNM 1772, pp. 5 - 8, 2002 © Springer-Verlag Berlin Heidelberg 2002
VX
+
(Jv)y.
Algebra
2 Linear
6
In this
case
(W, J)
and
split
as
a
we
(V, J)
call
(AJ
-JA)
=
Hom(V, W)
Hom+(V, W)
=
Hom(V, W)
and
If.
space.
(V, J)
maps from V to W complex linear (AJ = JA)
linear
homomorphisms.
Hom(V, W) fact, plication
(bi-)vector
complex quaternionic
a
such spaces, then the quaternionic direct vector sum of the real spaces of are
and anti-linear
In
Quaternions
the
over
ED Hom- (V,
complex
are
W)
vector
space with
multi-
given by
(x
+
iy)
Av
:
(Av)
=
x
+
(JAv)
y.
vector example of a quaternionic space is H" An example vector complex quaternionic, space is HI with J(a, b) := (-b, a). is simply left-multiplication On V H, any complex structure by some -1. lemma describes N E IHI with N2 The following that a situation for produces such an N, and that will become a standard situation naturally us. But, before stating that lemma, let us make a simple observation:
The standard
of
.
a
=
=
Remark 1. J E
for
End(U)
any
x
.0
On
a
2-dimensional
real
induces
vector
0. Wethen
compatible
J
call
space
the
following,
three
2
=
-
I
NU= : U U The pair 2.
(N, R)
is
=
and this
is
a
fx
E
HI
Definition normal
1.
as
real
unit
1V'
=
subspaces
U in the geometric
oriented
N, R (-= H
exist
=
2
(2.1)
R
(2.2)
UR, =
x}.
(2-3) there
is
only
one
normal -1
=
R,
vector
Ul of
of U
in Im H =
R3.
R2, the sets
:=IxEHINxR=-x}
dimension
2.
by (2) of the lemma, N and R are called a left of U, though in general they are not at all orthogonal
Motivated
vector
=
If U is oriented, up to sign. compatible with the orientation. above, and U C Im H, then
U:=jxEHjNxR=x},
right
structure
N is
Euclidean
orthogonal
Then there
unique
Given N, R E H with
are
2.
NxR
N=
3.
complex positively
properties: N
such pair such that If U, N and R are
is
0.
with
Lemma2 (Fundamental lemma). 1. Let U C H be a real subspace of dimension with
U each
(x, Jx)
0 such that
orientation
an
sense.
and to
Conformal
2.2
(of
the
to 1, then
a2
Proof
x
U\10}
E
(1).
I E U and if
If
(N, R)
1. Hence
-
easily from put U := x-'U.
follows
sign,
up to
lemma). =
then
1
NI
E U is
a
(a, -a)
=
a
works for
unit
U,
Clearly,
7
orthogonal
vector
and the
E U and Na E U. If
uniqueness, and arbitrary,
U is
U. Moreover,
1 E
Maps
(N, R)
0.
works for
Nx, R) basis of U, then R, and u, v is an orthonormal Use the geometric the requirements: N R u x v uv satifies properties of the cross product. of (3). The above argument shows that u(x) := NxR has I-eigenspaces 2. Since a is orthogonal, real dimension so are its eigenspaces.
only
U if and
(2).
If
(x
if
Example
=
=
Let
1.
mension 1. Then v
works for
U C Im H
=
=
-
and w, and
(V, J), (W, J) be complex quaternionic Hom+(V, W) is of real dimension 2. To
Then N2= -1 =
=7R'.
JF
FJv
-#=
But the set of all same
complex
such
result
vector
A linear
=
:
all
x, y
orthogonal
E V.
basis
"normalized"
Jw
=
Hom(V, W)
this,
of di-
choose bases
wN.
given by F(v)
is
=
wa, and
JFv
J(wa)
(Jw)a
=
dimension
Hom- (V,
and therefore
=
wNa 4=*
2, by the last
aR
part
=
Na.
of the lemma.
Hom(V, W) are W). As stated earlier, isomorphic with C. (non-canonically)
Maps
V
-+
Wbetween Euclidean
vector
spaces is called
conformal
A such that
positive
Fx,Fy
waR =
The
spaces
assume
Jv
FJ
vector see
This
equivalent
is
of V into
a
>=
A < x,y the
to
normalized
fact
>
that
orthogonal
F maps a normalized of F(V) C W. Here
basis
all vectors have the same length, possibly 54 1. C, and J : C -+ C denotes multiplication by the For x E C, IxI 4- 0, the vectors imaginary unit, then J is orthogonal. (x, Jx) form a normalized orthogonal basis. The map F is conformal if and only if On the other hand (Fx, JFx) is (Fx, FJx) is again normalized orthogonal. normalized if and only if Hence F is conformal, orthogonal. If
V
=
means
that
W= R'
=
FJ where the
sign depends
on
=
JF,
the orientation
behaviour
of F.
Algebra
2 Linear
involves
complex
the
for
is fundamental
If F
R'
:
condition
this
Note that
C
=
subspace
2-dimensional
of this
and injective,
H is R-linear
=
product,
scalar
the
generalization presented here. J. A
theory
R4
-+
involve
does not
structure
the
Quaternions
the
over
fact
to
U
then
but only quaternions
=
F(W)
is
a
H, oriented
of
by J. Let N, R E H be its left and Then NU U UR, and N induces an orthogonal right normal vectors. endomorphism of U compatible with the Euclidean scalar product of V. The real
=
map F
R'
:
U is conformal
-+
if and
conformal
exist
FJ
*F:=
This
leads
*df
df
:=
o
surface, TM-+
a
a
exist
N2= -1
*df
If f
is
an
Remark 2.
-
right
and
=
(2.4)
then
immersion
the left
called
functions
sense
f
(2.5)
C
:
conformal
-+
RI,
=
is
an
C,
maps into
i.e.
=
i.e.
a
:
C
-+
H is
such that
analog =
2-dimensional
-1. TM, j2 N, R : M -+ =
manifold
A map f : M H such that with
2
(2.4)
-dfR.
(2.5)
from (2.5), Of f.
vector
*df for
-I
R
=
Ndf
follows
normal
Equation
NF. Hence F
=
fundamental Riemann
Let M be
2'.
endowed with
R4 is
following
=
NF = -FR.
=
J: complex structure called conformal, if there J,
Definition H
the
to
only if FJ N, R E H, NI
if and
if there
only
=
and N and R
are
unique,
of
idf
In this Cauchy-Riemann equations. of generalization complex holomorphic
of the
H are
a
maps. -
-
real subspace. f is an immersion, then df (TpM) C H is a 2-dimensional there Lemma exist to a N, Hence, according 2, R, inducing complex strucThe definition that coincides with J ture J on TM1--- df (TpM). requires the complex structure already given on TpM. For an immersion f the existence of N : M -4 H such that *df Ndf for already implies that the immersion f : M-+ H is conformal. Similarly If
=
R. -
If
f
:
M-+ Im H
normal not
vector
orthogonal
of to
RI is
an
But for
df (TM).
then
immersion
general
f
:
M --+
N = R is
H,
the
"the
vectors
classical"
unit
N and R
are
3
Projective
Spaces
theory the Riemann sphere CP1 is more convenient as for than the complex plane. holomorphic functions target space Similarly, the natural for conformal immersions is HP1, rather than H. We target space therefore of the quaternionic give a description projective space. In
complex function
a
Projective
3.1
Spaces and Affine
The
quaternionic complex cousins,
tinuous)
projective as
canonical 7r
The manifold
defined,
is
the set of quaternionic
lines
in
7r(X)
=
similar
to
its
real
and
H1+1. We have the (con-
projection :
Ep+1 \f 01
__+
ff-lpn,
X
of Rpn is defined
structure
For any linear
HP'
space
Coordinates
form
P
(Hn+')
E
U:
+
as
[X]
=
XH.
follows:
P 54 0,
7r(x)
-+
_1
well-defined and maps the open set 17r (x) I < P, x > 54 0} onto the affine to Hn. Coordinates of this type are hyperplane P 1, which is isomorphic called affine coordinates for Hpn. They define a (real-analytic) atlas for ffffpn. often use this in the following We shall We choose for a basis setting: H1+' such that # is the last coordinate function. Then we get is
=
XJX-1 n+
X1
Xnxn+l
Xn
(Xnx*n+l) XlXn+1
or
I
i
Lxn+lj The set
17r(x) is called
the
hyperplane
at
I
=
01
3
10
Spaces
Projective
Example 2. In the special point: HP' is the one-point
case
n
1, the hyperplane
=
compactification
R4,
of
at
infinity
hence "the"
is a single 4-sphere:
Elp, that
however, 4-sphere, but Note
not
the on
unless
-
antipodal
of the
notion
HP'
we
map is
natural
additional
introduce
on
usual
the
like
structure,
a
metric. For space
our
it
purposes
Tilapn
for
For that
coordinates: h
=
#
If u o 7r
IT+'
:
\101
(Hn+')*
E
good description
a
purpose,
ffp+l
7:
in affine
to have
important
is
1 E ffffpn.
is
we
above,
ffP+l,
\10}
the
of the tangent projection
Hpn
_-,
as
consider
x
-+
then x
54 0
3.
IRpn.
For < v, (In the
Riemannian
and v,
H,
we
maps
j(p)
w
>=
1:
The corresponding conformal considerations. following
w
E
.,.
metric
on
on
Hpn
define
>=
have
d ,7r(v),
WX
>
< V'W >< X'X
>
< x,
>
x
>
>
< X'X
complex
product
inner
Pseudo-)
-
< X'X
on
(p), that,
by
Re
2
IX'W
>
(3-5)
.
ITkWk we obtain the standard Riemannian metthis is the so-called Fubini-Study metric.)
case,
structure
is
in
the
background
of all
of the
12
induces
Riemannian
standard
Wetake this it
Spaces
Projective
3
R4 via the affine
on
h
:
H2,
H -+
DIP'
t-+
1XI
S' and ask which metric
=
parameter
HP',x
and let
"=-"
(v)
(v0) (x)
:
(x, 1),
-+
x
on
H -+
h
Let
metric
_
equality
denote
(X)
mod
ff.
1
Then
6xh(v)((x)) The latter metric
d,,h(v)
1
is
vector
on
0
-orthogonal
.,.
VV)
'7-
xv
-
1 + xx -
(x, 1),
to
-
-,t
I I + xx-,
the
and therefore
induced
given by
H is
1
h*
>x
< v,w
stereographic
But
=_
=
(1
+
X.:t)3
(I
+
X;,-)
V
Re
R
Xj )2
+
>
W
the metric
4 < V5
(I+ Xj )2 on
R4. Hence the standard
Example 4. If
we
consider of
above construction but
these
points
metric
a
an
metric
< V,
lines
Isotropic
by
x
The
points -
is
E ImH
point
at
=
for
the
metric
on
lines
isotropic
case
n
=
4.
curvature
Hn+', (< 1,
then
the
1 >=
0),
1, and the hermitian
in
the
induced factor
metric -
affine
(X) (X) I
H is
0
S3
coordinates
h* < v,w
constant
(1)
infinity
3-sphere
a
in affine
ITIW2 + IT2W1
I
,
h
coordinates
x
:
-+
(xl)
by
,
+ X,
>= j
R3.
As in the previous
for
Hpn fails
W >=
characterized
are
0 =
R
0 of H. on
these
This
is
-
half-spaces.
up to
a
Two-Spheres
3.4
Moebius
3.3
Transformations
The group Gl(2, i.e. the set of all
4
acts
How is this
action
hermitian
metric
nite
IdG(d,,-x(vA))I'
=
=
compatible with the metric of V? Using (3-5) we find
G(O), G(vA)
"
Re
=
I
IA12
Re
Gv, Gv
Gx >2
Gx,
i.e.
the metric
for
is
a
conformal
induced
=
this
Two-Spheres
with
the
S4,
see
+
d)
I
+
is the full
of
b) (cx
+
=
cX
diffeomorphisms
complex
group of all
case.
orientation
preserving
confor-
[7].
S4
in
the set Z
-
Rj.
positive
given by
are
is known that
For S E Z
G(vA),
action,
p E
isomorphism. under by the pull-back 1 S4. We on RP GL(2, R acts conformally Moebius transformations In affine on Rp'.
constant
the
emphasises the analogy
We consider
a
I
d,,7r(vA)
+
3.4
Spaces
Projective
14
3
Proof
Weconsider
i.
H'
Then S is Clinear
as a
(right)
and has
complex
(complex)
a
S(vH) We choose
a
basis
and Sw = -vH
v,
w
affine
=
-1
E
=
h
parametrization
[vx +,w]
S'
:
R2,
IMP',
a
real-linear
equation
vNx
w)
+
Nx
this
thus the
Lemma2 this
is of real
-
-R
vN for
some
-
(vx
+
we
get:
w)-y
+
wR
w]
=
vx-y
+ w7
H = x-y -y
H.
for x, with
dimension
=
vH
-
Nx + xR
By
Sv
i.e.
[vx
x
Nx + xR is
S',
NH= HR.
3 S(vx
37
vH E
implies
-1
=
H -+
3.y
This
=
vN H = vH.
of EV such that
wR. Then S'
-
N2 For the
eigenvalue
imaginary unit vN, then
Sv
N. If
0.
Hence S'
N,
=
space with
vector
associated
=
homogeneuos equation
0.
2, and
any real
2-plane
can
be realized
way.
S and -S define Obviously, of the orientation an fixing lemma can be paraphrased Z is the
set
the
same
2-sphere. But S determines (N, R), 2-plane and thereby of S'. Hence
above real as
follows:
of oriented
2-spheres
in
S4
=
fflpl.
Bundles
4 Vector
We shall troduce
need vector
bundles
Quaternionic
4.1
the quaternions,
over
action
of H
vector
spaces.
Example the first
V from
on
The
5.
factor called
product
bundle
and the obvious the trivial
such
right
the
7r
that
a
-+
on
bundle
with _p
Zi
=
:=
1 (1, V)
:
Z
1. More
and vector
f*V:=
is
just
the obvious
the fibre
projection of V
f x}
on
x
Hn
E lffpn
X
Ep+1 I
_+
space structure
over
I(x,v)
IV
and vector
are
the
1-dimensional
precisely
E
Vf(,,)}
bundle
f (x).
F. E. Burstall et al.: LNM 1772, pp. 15 - 22, 2002 © Springer-Verlag Berlin Heidelberg 2002
V
C
1},
1. are
vector Example 7. If V -+ Mis a quaternionic 1 1 is a map, then the "pull-back" f *V 4Mis
with
projection
Hpn
___,
Rpn, (1, V)
7rz
The differentiable
the
each fibre
bundle.
7r_,
line
fibre-preserving
become quaternionic
M with
space structure
vector
smooth manifold
a
smooth
fibres
the
M x Hn
:
n over
Example 6. The points of the projective space Elpn bundle subspaces of Hn+'. The tautological
is the
in-
Bundles
Vector
A quaternionic bundle 7r : V -+ Mof rank vector with Mis a real vector bundle of rank 4n together
is also
briefly
and therefore
them.
the
bundle defined
obvious over
ones.
M, and f
:
M
by
CMX V structure.
The fibre
over
x
E M
Bundles
4 Vector
16
maps f the associate
be concerned
We shall
projective space. x is f (x) c ffP+l product bundle
f
To
with
we
JxJ
=
M -4, RP'
:
bundle
L
H1+1. The bundle
x
L is
an
Maps
All tions
natural
bundle
vector
H induces
quaternionic
two
all
x
the
Over lffpn
Example
8.
inside
the
vector
spaces
bundles.
For
A section It
is
maps
of isomorphism
notion
it,
V2 ).
4ilv,.
restriction
obvious
!P E
fibre-wise, a
to
subbundle
F(Hom(Vi, V2)) V,
:
V1,; homomorphically
for
vector
M
operaL of a
--
into
is called
a vec-
V2 such that for V2x. There is an
bundles.
product
bundle
H
=
HP'
x
H1+1 and,
Z. Then
subbundle
Hom(Z, HIZ),
Let L be a (and Definition). -+ E HIL -P(Hom(H, HIL)) 7rL .P(L) C F(H) is a particular map dO(X) E Hp Hn+', and 9
:
subbundle
line
H
be the M
-+
of H
projection. Hn+1
.
If
=
(do (X))
WL
E
(HIL)p
=
X E
EP+1 ILp.
A: M-+ R Then 7rL
(d(0A) (X))
7rL
=
(dO(X)I\
+
Odl\(X))
:--:
Wesee that
0 is tensorial
in
0,
i.e.
we
J(X)
-+
7rL
(4 (X))
:;--:
6 (X)
(0)
obtain :--
JL (X)
G
Hom(Lp, (HIL)p).
7rL
Mx Hn+'.
A section
=
Let
:
(3.3).
Example
Let
f
Mx Hn+1
=
extend, example,
smooth map !P
have the
we
tautological
a
THP' see
map
a
bundle HIL with fibres Given quotient H,;IL_-. V1, V2 the real vector bundle Hom(V1, V2)
a
homomorphism.
bundle
over
of the
subbundle
bundles
vector
Hom(Vi.,
has the fibres tor
for
of vector
category
the
subbundles
Line L c H
constructions
the
in
line
into
identification
M_+ ffffpn
:
f Z,
a
surface
whose fibre
L of H over Mdetermines
HP'
f
a
*
EP+l.
H:= Mx
Conversely, every line subbundle by f (x) := L,,. We obtain
:
from
=
(dO(X))I\-
TpM,
0
c
then
Quaternionic
4.1
Of
is R-linear
this
course
values
Mwith
in X
us
0
F(L)
E
Given
repeat:
p E
0(p)
such that
=
I Jp(X),Oo similarity
Note the
=
E
submanifold
of
a
to
TM and H1+1
M, X E TpM, and 00 Oo. Then
7rL(dv0(X))
clearly comparison
Euclidan
the
change
=
L
with
of
there
is
a
section
m
form
.
In the
case
at
hand,
bundle.
normal
subbundle
a
as
a
map
f
M -+ HP'.
:
L in
a
corresponds the general
L
This
is
connected)
(covariantly this
Even if
the second fundamental
to do with
is
an
form of
immersion, f. Instead,
I shows that
Proposition
Hom(L, HIL)
6: TM-+
corresponds of L
on
1
(dY (X))
the
's to
Lp,
E
dpV)(X)
=
space.
A correspond
nothing
has
1-form
H.
We can view 6
as a
(4.1)
the second fundamental
to
Min
measure
bundle
vector
be viewed
17
f2l (Hom(L, HI
a(X, Y)
method to
6 should
so
Bundles
Hom(L, HIL):
in
IJ Let
well,
as
Vector
to the derivative
of
f
and
,
we
shall
therefore
call
it the
derivative
.
Example
10.
The dual
space V is,
vector
quaternionic structure:
in
vector
For
w
E
a
V`
:=
natural
spaces
fw
-4
left be right
way, to
V
:
vector
extends
rank
1, then L*
to
space. spaces,
of
quaternionic
a
But since
the
we use
we
choose
opposite
V* and A E H we define
w.
wA := This
Hjw H-linearl
H-vector
a
quaternionic
vector
is another
quaterionic
A quaternionic vector product bundle Mx H,
bundle
bundles. line
is called
E.g., if L is a line bundle, i.e. of bundle, usually denoted by'L-1. trivial
if it
with the isomorphic M-+ V : global 0,, that form a basis of the fibre line everywhere. Note that for a quaterniQnic bundle over a surface the total space V has real dimension 2+4 6, and hence 0 : M -+ V has codimension 4. It follows from transversality any section deformed so that it will not hit the theory that any section can be slightly Therefore there exists nowhere vanishing 0-section. section: a global Any line bundle over a Riemann surface is (topologically) trivial. quaternionic i.e.
if there
exist
sections
is
,
=
4 Vector
18
Bundles
Complex Quaternionic
4.2
complex quaternionic
A
nionic
bundle
vector
bundle
vector
V and
Bundles is
a
J E
section
a
pair (V, J) consisting with 1'(End(V))
of
a
quater-
j2 section
see
2.1.
Example L
=
Given
11.
Mx H has
a
Example 12. For
f : M -+ H, *df Ndf the quaternionic complex structure given by =
a
given S
E
S'
Jv:=
Nv.
End(H2)
with
=
111
Sl
=
line
,
1}
S2 C
=
-1,
we
bundle
identified
HP,
in RP1, see. section 3.4. Wenow compute J, or rather the image line bundle L. In other words, we compute the corresponding tangent space of S' C HP'. Note that, because of SL C L, S induces a complex structure on L, and it also induces one (again denoted by S) on HIL such that irLS S7rL. Now for V) E r(L), we have as a
of
6,
2-sphere for
the
=
6SO This
=
7rLd(SO)
real
But the
S' is
vector
an
For
our
Lemma3.
image 6
a
example
=
S60.
U is Note:
-R rather
C
is
an
quaternionic
we
real
End(V),
then
2,
see
Example 1,
and
equality:
Hom(LI, (HIL)j)
V, W'be 1-dimensional
oriented,
the
has rank
Lemma2.
2-dimensional J E
Hom+(L, HIL).
generalize
next
U
with
S7rLdo
the inclusion
=
vector
TjHP1.
spaces,
and
Hom(V, W)
subspace. Then there exists a pair of complex End(W), unique up to sign, such that
vector
j
E
ju
If
=
Hom+(L, HIL)
surface,
UC
structures
C
Hom+(LI, (HIL)I)
=
Let
=
bundle
embedded
TIS'
be
7rLSdV)
shows
TS'
since
=
=
fF
there
E
is
=
U=
Ui,
Hom(V, W) I jFJ only
one
=
such pair
-Fj such that
J is
compatible
orientation.
Here
than
we
R.
choose
the sign
of
J in
such
a
way that
it"corresponds
to
Complex Quaternionic
4.2
Proof.
Choose
Hom(V, W) 1
The
EIPI
basis vectors v E V, w E W. Then elements endomorphisms of V or of Ware represented by quaternionic
and therefore
following
Proposition with
is
now
HIL,
M x W be
=
Definition
with
A line
3.
the
J
on
=
HPI, is
a
if
i.e.
real
subbundle
1'(End(H/L))
f
Let
13.
normal
E.F(L), J(
:
we
define
=
by J: T ,M
M x Hn-- '
over
in
curve
-+
a
6(TxM).
Riemann
Rpn, if there
sur-
exists
a
6j.
L is
2,
immersed
an
such there
then
also
is
holomorphic
J(TM)
that
a
C
complex
curve
in
Hom(L, HIL) structure
J E
(f) I
J E
(4.2)
jj
=
if and only into HP1 is a holomorphic curve with the are compatible given by. the proposition
on
M-
vector
Min the H be
R, and
a
let
sense
of
(4.2).
conformally L be the
M-*
immersed line
bundle
Riemann surface
corresponding
HPI.
and
R)
=
=
If
=
If
:
Then
E M
immersed
complex structures given complex structure the
Example right
x
such that
A Riemann surface
with
induced
injective,
of rank
*6
if
all
ji,
L C H
we see:
addition
in
for
L such that
proposition 6 is
surface in unique complex
oriented exist
6(Txm)j,
holomorphic
a
*6 From the
=
orientation
or
immersed
such that
6(T,:M)
subbundle
conformal
face M is called complex structure
of Lemma2.
that
to
Then there
by J, j,
J6
compatible
an
(Hom(L, HIL)). denoted
h(TXM)
and J is
reduces
in
evident:
5 E J?I
L and
on
the assertion
Let L C H
3.
derivative
structures
19
non-zero
and
1-matrices,
x
Bundles
7rLd((fl) d '") 7rL
End(L) by
0
J
R) =
7rL((dof) (*Of)d
R+
-7rL
R then
(fl)
dR)
(fl)
to
Bundles
4 Vector
20
jj
(L, J)
hence
is
=
1
right
normal
4.3
Holomorphic
Let
(V, J)
I
some
Quaternionic
complex quaternionic
a
R
is
holomorphic
a
is conformal
E and f
M-+
:
(L, J)
if
with
R.
vector
be
R for
-
*J,
Conversely,
curve.
M M
J
curve,then
holomorphic
a
=
Bundles bundle
vector
the Riemann surface
over
M. We decompose
HomR(TM, V)
KV (D
kV,
where
jw:
KV:=
jw:
KV:=
Definition
A
4.
TM
V
TM-+ V
holomorphic
structure
I
I
* w
* w
on
=
Jwj,
-Jwl.
=
(V, J)
is
linear
quaternionic
a
map
D
for
such that
0
all
1'(V)
E
0 E.P(V)
is
=
(DO)A
called
valued
-1 (dA
1.
For
+ i
(dA
*
a
dA).
+ i
*
In
only
(D,O)A 2.
+
=
if Do
=
*
dA).
(4.3)
0, and
we
put
ker D C F (V).
part
of
this,
k-part)
(the
note
for
that
of dA is
given
complexby OA
fact, =
structure
is
way to make
OOA".
1(OdA + JO
2
understanding
dA) (JX)
A holomorphic natural
better
anti-C4inear
A the
+
holomorphic
HO(V) Remark 3.
P (EV)
-4
and A: M-+ H
D(OA) A section
P (V)
:
*dA(X) a
-
i
dA(X)
=
-i(dA
generalized 0-operator. of a product rule
sense
+ i
*
dA) (X).
Equation of the form
(4.3) is "D(,OA)
the
in ELI"', does this mean L carries curve a natural holomorphic structure? This is not holomorphic yet clear, but we shall come back to See also Theorem I below. this question.
If
L is
a
Holomorphic
4.3
Quaternionic
-1, Example 14. Any given J E End(H), j2 Then F(H) bundle. vector complex quaternionic
turns
=
1(do
D,O is
holomorphic
a
+ J
2
H
Bundles
=
21
M x H'
H},
M-4
into
a
and
do)
*
structure.
line bundle and 0 E F(L) has no complex quaternionic D on (L, J) such exactly one holomorphic structure In fact, any 0 E r(L) that 0 becomes holomorphic. can be written as'O Op with p : M-* H, and our only chance is
Example
If L is
15.
then
zeros,
there
a
exists
=
1
Do This,
indeed,
Example that
2
(Odp of
the definition
+
(4.4)
JO * dy).
holomorphic
a
structure.
f : M-+ H is a conformal surface with left normal vector N, Mx H, and there exists a unique D such for L a complex structure A is 0. section 0 holomorphic if and only if dp + N * dp lp 0, If
16.
then N is D1
satisfies
:=
=
=
=
=
i.e.
*dl-t holomorphic
The left
normal
sections
f
N as
are
In this
.
case
=
Ndp.
therefore
the
dim HO(L)
! 2,
since
maps with the same I and f are independent
HO(L).
in
Theorem 1.
by
Jw
:=
wJ.
characterized H induces all
If
a
the
The
by
the
section
M x H'+1
with complex curve holomorphic structure a complex defined D pair (L-', J) has a canonical holomorphic structure linear form W : Hn+1 -4 following fact: Any quaternionic to the fibres by restriction WL E V(L-') of L. Then for
L C H
then
J,
structure
=
dual
L-1
bundle
is
a
inherits
w
DWL::::::
Proof. a
conformal
The vector
total
has
bundle
space of real
DWL
15
yields
0. Now any a E F(L-') by (4.4), for any section
=
2
+
>)
WA, J'O >)
JO >)
have
1 -
2
I -
2
4 Vector
22
*J
Note that wA by
=
such that
As a
this role
term
< a,
reference no zero.
we
shall
is
+
*d(JV))
E
F(L),
and this
allows
us
to replace
well:
> +
to
*
d < a,
w, hence
But the last
JO >)
D is
equality
-
2
< a,
independent shows Da
do
+
of the =
0
for
*dJO
>
choice any
a
of =
W
WL
the next section, see in L in a holomorphic curve In higher dimensional structure. holomorphic projecL-1 rather than L plays a the case. Therefore no longer higher codimension.
natural in
do as
Sphere
5 The Mean Curvature
S-Theory
5.1
Mbe a'Riemann
Let
surface.
Let H:= Mx
denote to
product
the
S2
with
-1 be
=
a
bundle
M, and let S: M-+ End(EV) E F(End(H)) the differential on H. Wesplit according
over
complex
19
structure
type:
do where cP and d'
=
denote the Glinear
*d'
do
d"O,
and anti-linear
Sd,
=
+
*d"
=
components,
respectively:
-Sd".
Explicitly,
do So d" is
(do
-
d"O
do),
S*
2
(do
+ S
while d' is structure on (H, S), holomorphic of (H, -S). a holomorphic structure general d(SO) i4 Sdo, and we decompose further:
structure, In
2
a
*
dio).
an
anti-holomorphic
i.e.
d=,9+A,
d"=a+Q,
where
a(SO) AS For
example,
we
explicitly
S90'
5(so)
-SA,
QS
I
on
0 defines
H, while
a
Sao,
-SQ.
have
2
Then
=
=
(d"O
holomorphic structure Q are tensorial:
Sd'(So)).
-
and 0
A and
F. E. Burstall et al.: LNM 1772, pp. 23 - 30, 2002 © Springer-Verlag Berlin Heidelberg 2002
an
anti-holomorphic
structure
Sphere
5 The Mean Curvature
24
QEI(kEnd-(H)).
AEr(KEnd-(H)), For
M-+
(dS)O
IV
E F
(H)
have, by definition
we
=
d(SO)
=
(,g
=
ASO+
QSo
=
-2S(Q
+
=
2(*Q
*A)O.
+
dS,
+
A)O
SdO
-
A)SO + (5
-
of
(5.1)
Q)SO
+
SAO
-
S(o9
-
-
S(5
+
Q)0
SQO
-
A)O
Hence
dS
=
2(*Q
*A),
-
*dS
2(A
=
-
Q).
(5.2)
*dS).
(5-3)
Then
SdS
Q Since
Remark 5.
Q 0. If dS S decompose H =
deviation
5.2
+
A),
(SdS
4
*dS),
-
(M
=
x
C)
"complex
L and
an
E
on
HIL
(SdS
+
ED
(M
x
C).
Therefore
A and
immersed
the
measure
Sphere
holomorphic
HIL)).
HP1 with deriva-
L C H in
curve
Then there
exist
complex
structures
J
such that
We want to extend
J and
j
=
to
S E
a
jj
complex
structure
F(End(H))
such that
SL this
Q
case".
*J
Note that
=
=
S?'(Hom(L,
JL
4
0 if and only if A 0 Q are of different type, dS of the complex endomorphism 0, then the i-eigenspaces
The Mean Curvature
J
A
A and =
from the
Wenow consider tive
2(Q
conversely
whence
and
=
implies
=
L,
SIL
=
Ji
7rS
=
j7r-
of
H,
i.e.
find
an
on
7rdS(O)
7r(d(SO)
=
SdO)
-
Sphere
The Mean Curvature
5.2
JJO
=
ho
-
=
25
0,
and therefore
(5.4)
dSL c L. of S is'clear:
The existence bundle
L'
Since
L'
Identify not unique,
L. is
H
=
0,
R can be
Q:
interpreted
I((S
4
+
1(SdS 4 Q+ V)
E
F(L),
SIL
complementary Ji SIP := j.
:=
S + R is
kerR,
and
We compute
If
some
if and
RS + SR Note that
for
and define
7r,
RH c L c whence R2
L E) L'
=
It is easy to see that S is not unique. if M-+ End(H) satisfies R: only
such extension
another
Write
HIL using
with
1
4
+
*dS)
+
-
element
as an
R)d(S
(SdR
0.
=
R) I
*d(S
-
(SdR
4
of
Hom(H/L, L).
+
R))
+ RdS + RdR
+ RdS + RdR
-
-
Then RV
*dR)
*dR).
then 0
=
d(RO)
=
RdRO
=
dRO + Rdo,
-R 2do
=
0
and, by (5.4), R
We can therefore
=
0
continue I
00=QO+ (SdRO 4 QO+
dSO
I
4
-
(-SRJO
*dRO) + R*
=
JO)
QO+ =
1
4
(-SRdo
Q0 +
+
(-SRJO
*Rdo) +
Rj =RS=-SR
Hence, for 0
E
F(L),
00
=
Q0
-
-SRJO. 2
JO).
R.
5 The Mean Curvature
26
with
Now we start
54
-2SQ(X)6(X)-'7r:
0. First
note
of X
In
fact,
F-+
c
=
sinO
some
X
54 0. cosO, s
=
Q(cX
+
X
SJX) (J(CX
that
this
H
definition
SJX)) -')
is
Q(X) (CI
=
in view
+
of
(5.5),
define
(5.6)
H
-+
positive-homegeneous
R is
+
(J, J) and,
S of
any extension
R= for
Sphere
of the choice independent of degree 0, and with
SS) (J(X) (cl
+
ss))
-1
Q(X)6(X)-'. Next
RS
=
=
=
By
-2SQ(X)6X-'7rS -2SQ(X)SJj1-7r
=
=
-SR.
(5.6)
definition
kerR,
L c
and from
-2SQ(X)6x-lj7r 2S2Q(X)6 X1jr
(5.3)
and
(5.4)
we
L D
get
I(SdS
*dS)L
-
4
=
QL,
whence RH c L. Wehave
(5.5),
we
find
now
for
00
shown that
0 =
=
This
S + R is another
extension.
Finally,
using
F(L)
Q0 QV)
-
-
-1SRdo 2
=
QJ-lirdo
Q0
I -
2
S(-2SQ6-'7r)do
0.
=
shows
Theorem 2.
HP'.
E
Let L C H
Then there
exists
a
=
M x EV be
complex
unique SL *6
a
=
=
L, 6
holomorphic structure
dSL C 0
S
QJL
=
=
0-
S
0
S
curve on
immersed
into
H such that
L,
(5.7)
6,
(5-8) (5.9)
Hopf
5.3
S is
SpLp
a
terms. 2-spheres, a sphere congruence in classical sphere Sp goes through Lp E HP', while dSL JS S6) implies it is tangent to L in p, see examples
In
an
affine
coordinate vector
Definition
f
as
the
mean
Equations 0 + Q
Remark 6.
whence d"
=
holomorphic particular,
a
Example 17.
holomorphic S E
Let
2-sphere
KPI.
in
11
S2 E
complex sphere congruence of L by definition, S: Wehave SL and Q 0. !(SdS *dS) 4 structure
the
In the
shall
we
definition
of the w
can or
be
for
Hence
C
0
F(L),
E
immersed
an
and,
in
HP1
corresponding
line
bundle
from the immersion.
and endow
Then the
mean
map S' -+ Z of value
constant
dSL
implies
A
encounter differential frequently wedge product of 1-forms
O(X, Y)
generalized verbatim to Vi, provided there
bundles
End(V)
composition
F(L)
E
10}
=
C L
Fields
following
the usual
The
L.
=
-
Hppf
11
=
and the constancy
=
5.3
sphere (congruence) of the Hopf fields of L.
simply the
L is
the
motivates
Then
-I.
HP1 I Sl
inherited
curvature
=
=
Let L denote
the
same mean
subbundle of (H, S, d") a holomorphic bundle itself. vector quaternionic
End(H2), S'=
S' with
9 and 12.
called
are
(5-7), (5.8) imply do + S * do 12 (d + S * d) leaves L invariant.
HP is
in
curve
is
=
Remark 9. This
see
curvature
forms A, Q E S?'(End(H))
differential
a
H at p,
=
sphere Sp has the
L the
=
1
M-4 R4
:
S is called
5.
I,]
system
(or,
C L
=
curvature
is
Because
the
equivalently,
27
of
family
Lp
=
Fields
x
w(X)O(Y)
=
a
End(V)
-+
product
(Vi)
with
V,
x
or
the
End(V)
Note that
w(Y)O(X)
-
forms wi E 01 is
forms.
V2
-+
values V.
in vector
Examples between
pairing
spaces are
the
the
dual
V* and V.
On
a
use
the latter.
form As
be written
any 2-form
o-(X, JX)
an
example,
A
O(X, JX)
w
will
M,
Riemann surface
by the quadratic
=
=:
u(X),
a
E
and
w(X)O(JX)
-
S22 is completely we
shall,
for
determined
simplicity,
w(JX)O(X)
as
(5.10)
wAO=w*0-*wO. We now collect curvature
often
sphere
some
information
congruence
S
:
about
M-+ Z.
the
Hopf fields
and the
mean
Sphere
5 The Mean Curvature
28
Lemma4.
d(A+Q) Proof.
(5.2)
from
Recall
2(QA Q +AAA).
=
SdS
using AS
Therefore,
d(A
-SA, QS
=
Q)
+
=
2(A
=
+
Q).
(dS
A
-SQ,
=
Id(SdS) 2
1 =
2
dS)
Q) =2(AAA+AAQ+QAA+QAQ). =
But A A
Q
0
=
by
2S(A
following
the
QA A
Similarly stabilizing
Notice
that
QxO 0 QzO independent if
=
=
Proof. which
=
A*
=
ff",
Q
*AQ
-
0, because A
Let L C H be
Lemma5.
7r
A
S(A
+
Using
and
that
k))
Q "left
have
AAQ
H
Q)
type argument:
"right
A is we
+
L such that
an
A(-SQ)
=
is left
K and
Q is right
surface
immersed
dSL C L.
Then
(-AS)Q
-
QJL
and S =
0 is
=
0.
(5-11)
K.
a complex equivalent
structure
on
to AH c L.
images of the 1-forms A and Q are well-defined: also QjXO -SQxO 0, and thus for any Z E TM. In other words, the kernels of Q and A are of X E TM. The same remark holds for the respective images. the kernels
0 for
Wefirst stabilize
(dw (X, Y) 0)
and
X E TMthen
some
need
formula
a
=7r
for the derivative
wL C L. If
L, i.e.,
(d(wo) (X, Y)
=7r(x
-
=
(W(Y)O)
7r
+ -
=
w
Y
*7rL,
A -
then
=
of 1-forms for
0
E
w
E
Q'(End(H))
F(L)
dV) (X, Y))
(W(X)O)
-
W([X, YDO %vo
.11
EF(L) +
w
(X) do (Y)
=6(X)w(Y)O =6(X)w(Y),O =(J A w + -7rw wedge
-
w
(Y) do (X))
J(Y)w(X)O
7rw(X)dO(Y) J(Y)w(X)O +,7rw(X)JO(Y) A J)(X, Y)O, -
-
+
-
-
7rw(Y)dO(X) 7rw(Y)JO(X)
Note that the composition 7rWJ makes composition. L, and L is annihilated by 7r. We apply this to A and Q. Since AL C L, QL C L we have on L, by lemma 4,
where sense,
we
because
w
over
(L)
C
The Conformal
5.4
0
I7r(QA
=
similar
type argument
a
to
(5-11),
7rA A 6
similarly
and
the
for
remaining
=
7rA
=
-2S7rA6,
Since
AL C L and 0
5.4
=
QJL
L
:
For
6.
End(V)
-+
quaternionic
a
Example 18.
of
is taken
indefinite
an
for
0
=
-7rQ
Further,
A 6.
--xSA6
=
SJQIL
or
-
-
X
54
0 is
an
isomorphism,
we
get
n
and
Gauss Map vector
A >:
:=< AB >.
the real
scalar
(a)
=
with
=
4
ja2
+
=
+
ka3
E H
we
< I
>= 1.
have
ao,
and
=
,
=
sur-
0,
conformal.
proposition,
S is also
called
the
conformal
Gauss map,
see
30
5 The Mean Curvature
Proof.
Wehave
QA
=
Sphere
0, and therefore
=
=
(5.12)
0.
Then, from (5.2),
=4
< -S(Q
+
Q
A, Q
=4
=
= 4
>
Similarly,
=4
QA >
+
=0
But,
by
a
of the real
property
=
=< -SQA
SQQ>
=
=< -SQQ >= 0,
SAA >
=
=< -SAA
>=
0,
>= 0.
-
).
Surfaces
6 Willmore
Throughout
The
6.1
this
Mdenotes
section
Energy
a
compact surface.
Functional
The set
IS
E
RP1 is
a
Z
of oriented
2-spheres
in
TsZ
=
-Z
=
S
Here
we use
the
in
Section
Definition
surface
A,
B >:=
=
-
8
traceR(AB)
functional
of
a
map S
:
M --*
Z
Of
a
Riemann
by
S of this
maps
_jj
5.3.
E(S)
harmonic
submanifold
=
E
inner
The energy
7.
M is
End(V) I S2
Fnd(EV) I XS -SX}, I YS SY}. JY E End(fff)
IX
(indefinite)
.
with
respect
to
variations
of S
are
called
M to Z.
S is harmonic
if
and
only if the Z-tangential
component of
dS vanishes:
(d This
condition
is
equivalent
to
*
any
dS)T of
the
F. E. Burstall et al.: LNM 1772, pp. 31 - 38, 2002 © Springer-Verlag Berlin Heidelberg 2002
=
0.
following:
(6-1)
Surfaces
6 Willmore
32
d(S
In
dS)
*
d
*
A
d
*
Q
A
=
=
o,
(6.2)
=
0,
(6-3)
=
0.
(6.4)
fact,
d(S Proof.
St be
Let
a
Q
*
=
4d
*
of S in
variation
fm
d
d
Wt- E(S) the
4d
=
S(d
dS)T
*
Z with
=
(Sd
variational
*
dS)T.
(6-5)
field
vector
Y.
-YS and
Then SY
Using
dS)
*
Wtwedge
=
(5.10)
formula
< dS A *dY > =
+ < dS A *dY
traceR(AB)
and
dS(-dY)
*dS
-
traceR(BA),
=
we
dY >=< dY A *dS >
*
>.
get .
Thus d
Wt- E(S)
2
=
Therefore
f'
< dY A
*dS
>= -2
JM
S is harmonic
For the other 0
only
if and
first
equivalences, =
d
=
(d
d(S')
* *
with
*Q
-
8d
*
*
dS >= -2
*
fm
.
dS is normal.
note
+ S
2(*dS)2
2dS A *dS +
Now, together
if d
Yd
dS)
*
*dS A dS + dS A *dS + Sd * dS
-
-
=
4(
+
),
(6-6)
dS A SdS >=
4(
Q A *Q >).
(6.7)
-
0.
>O.
(6.9)
Surfaces
6 Willmore
34
Proof, *A >
< AA
traceR(-A'
8
(*A)'
-
4
2 traceR A
=-ASSA=A2
Because dim L
2
=
we
2 traceR A
proves
(6.8).
Proposition
6.
This
The
=
is
closed.
If
S
:
Z, and dS
M-*
S*w In
=
=
2-form
-
w
E
S?2 (Z)
defined
by
forSEZ,X,YETsZ,
*A)
usual,
as
2 < A A *A > -2
see
Q A *Q
.
particular, 1
degS:=
7r
is
2(*Q
IL7
from Lemma7.
ws(X,Y)=<X,SY>,
2.
12
traceR A 12L
(alternating!)
The
1.
2
have
follows
positivity
A
traceR
-
a
topological
f
- A M
of S.
invariant
S maps the surface Minto the 8-dimensional Z, deg S cermapping degree of S. But for immersed holomorphic curves of two mapping degrees deg S it is the difference deg N deg R, where N, R : M-4 S2 are the left and right normal vector in affine coordinates, see chapter 7. Remark 7.
tainly
Since
is not the
=
Proof.
(i).
the 2-form
Weconsider
COS(X, Y) Then
dsCo (X, Y, Z) is
a
1
:=
linear
-(< X, 2
X, Y, Z
E
TsZ,
S E Z,
End(EP) SY
>
combination
(< dS(X)SdS(Y)
>
< dS A
-
-
)
)
(X, Y),
SdS >
yields the formula. under deformations invarlance topological 3 deforms So: M-4 If 9 : Mx [0, 1]
and Lemma6 The
theorem:
fmx[0
0
f fm
6.2
that
integral
=
4
=
8
fm fm
g*W
g*"' J MX0
S*1W-
+
A
*Q
>
problems the
variational
energy
-
A A *A
)
invariant
functional
can
be
replaced
by
of < A A *A >.
The Willmore
Definition
Hopf field
Stokes
1]
topological
the
from
then
From
E(S)
we see
S1,
dg*w ,
MAx 1
Remark 8.
of S follows 3 into
8.
A.
Let L be
Functional a
The Willmore
compact immersed holomorphic
functional
W(L)
:=
-I 7r
f
A M
defined
of
L is
as
curve
in
HP' with
36
Surfaces
6 Willmore
If we vary holomorphic
the
J
structure
Willmore not
On the
on
Msuch that
3.
Critical
Proposition
this
treat
HP, it will in general not hand, any immersion induces
M --+
:
other
with
Wwith
called
Willmore
Proof. Lt
if
(Ejiri and
[2],
curve,
of
fixing surfaces,
L
Willmore
the
R4
in
conformal
but
we
As usual,
we
7r, o
its
a
2
(H
K
-
have
we
Kj-) ldf I 2
-
An immersed
holomorphic
sphere S
is harmonic.
curvature
mean
variational
abbreviate
=
7r(SO)*
dt
-
irS
d
structure
d
E(St)
dt
t=0
=
general
fm
on
t=o f
+
>=< B A *A
fm >,
+
because
Hence
fm
traceR(AB)
< dS A
=
*d
>.
traceR(BA).
(6-11) Next
we
claim II
=
0.
(6-12)
The Willmore
6.2
B,
On TMlet
w(X)
i.e.
(X, JX)
conformal,
see
dS(X) dS(JX)
=
= 0
I.
,d*dS
< ,Sd*Q>
(, Sd
traceR
*
Q).
A M
lemma that
following
consider
we can
-
dS(Y)dS(JX)
=
imaged Therefore
-
A M
2
show in the
dS(JX)MS(X) > > < dS(JX)dS(BX) < dS(BX)dS(JX) >
> +
we
dS(X)dS(JBX)
*dN,
NdN >)
(7.7)
-
_III(X,
jX)12.
Surfaces
7.1
4KIdf 14
=
-
Using (7.1)
K.
RdR, RdR >
-
(7.5)
corollary
*
dNdf,
computation,
df (*dR
+
>
+
>
NdNdf
+
> >
dfdR, N dNdf
> +
dNdf
+
*dNdf)
dR +
*
N(-dfdR
dNdf, dfR
+
-df
dR +
41
>
*dNdf
dR +
*
*dNdf),
dNdf, N(-df
dR +
*dNdf
df, -df dR + dNdf
*
dNdf ), -df
dR +
*
dfdR
+
M-4
-dR +. *dRR
R, this
1 =
16
JRdR
is the
-dR
--
integrand -
*dR12
classical
< A A *A >=
-
is
1 =
4
R * dR
-
integrand 1
4
(Ih 12
-
=
-v.
by
given
(IHI2
*dN).
Coordinates
viate
=O=<W'W>'
Surfaces
10.1 Let at
L be the
an
adjoint
stabilizes
Lj-,
S3
in
bundle
line
isotropic
map M -*
and L
=l.
=
L
Z7p 1
with
SP*
-+
mean
with
to
implies S*L
S*L'
=
=
L'
=
L.
Similarly,
(dS*)L Moreover,
if
Qt belongs
to
S*,
1
(dS)*L
=
C L
then
Qt
(S*dS* 4 -
(dSS
I(SdS
4
-
-
*dS*)
*dS)* +
-A*.
F. E. Burstall et al.: LNM 1772, pp. 61 - 66, 2002 © Springer-Verlag Berlin Heidelberg 2002
sphere
curvature
respect
*dS)*
L.
.
S. We look
Clearly
S*
10 Willmore
62
kerQt
Therefore
We proceed
S*
=
S3
in
(image(Qt)*)'
=
sphere,
curvature
mean
(imageA)-L
=
S and S*
show that
to
of the
uniqueness
Surfaces
L'
D
coincide
on
L and
HIL.
By
Theorem 2, it then follows
see
the that
S.
Let
0
E
1'(L),
and write
SO
=
OA,
S*O
0,60
>=
Op
and
= 0. Differentiation
of
0.
obtain
JO, SO>+< 0,(dS)O
0 =
+
I.,
=0
=
=
A+ P
=
Now we
apply
*
+
+
u.
>
0, SJO >
= XW.
>= WP
10.2
Comparison
(10.2)
with
shows p 0
It
follows
a
A
=
& W + WA
=
SIHIL
i.e.
p,
=
A, and
=
63
&)WA.
-
S*IHIL.
=
2-Planes
get
we
(
=
Hyperbolic
completes
This
the
assumptions
of Theorem 2, and S* = S by uniqueness. if S* = S and So OA, then Conversely, =
0,0
A < Now S2
>=
=< W,
v
W>=
0,
< V,
W
Then
S N2
with
=
R2
1, NH
=
(0
N -H
=
-R
HR, and S'
is the
C Ifff
locus
of
Nx+xR=H. If S' is invariant
iR
=
H
under the reflexion
at
S3,
is the locus
also
of -Nd
-
or
Rx + xN
According
then it
to section
=
T1.
(H, N, R)
3.4, the triple
unique
is
up to
sign. This implies
either
(H, N, R) By (10.1) intersects
either
S*
orthogonally,
=
=
(ft, R, N) S,
and the
and S*
=
(H, N, R)
or
2-sphere
-S.
lies
-R, -N). within
We summarize:
the
3-sphere,
or
it
64
Proposition termined by S*
=
in
S3
A 2-sphere S E Z inner product indefinite
the
intersects
19. an
hyperbolic 2-planes
hyperbolic
in
de-
4-spaces
if
and
only if
-S.
10.3
Willmore
Hyperbolic Let L be set
Surfaces
10 Willmore
of
Surfaces
Surfaces.
in
4-Space Willmore
connected
a
indefinite
an
S' and Minimal
in
surface
hermitian
form
S3
in
C
HP',
where S3 is the
Then its
H.
on
mean
curvature
isotropic sphere
satisfies
S* Let
us
assume
B.icklund
A
that
transforms
$ 0,
and let
S.
L
ker A and
L
L.
image Q be
the
2-step
of L.
Lemma13.
Proof.
First
have
we
I
Q*=
4 I
4
(SdS
(-SdS
imap Q is S-stable,
Now
Therefore
=
L'
=
=
-
*dS)*
*dS)
-
and S* on a
4
=
=
(dSS
-
-A.
S and
(10-3) So
=
dense open subset
(image Q)'
=
*dS)
kerQ*
OA imply
= 0.
of M
ker A
=
Lemma 14.
-S
for
the
Proof. and
mean
First
sphere 9 of L.
curvature
L
=
L
is
Q and, therefore,
under I
4
and this
vanishes
sphere by
these
on
three
(-S)-stable. d(-S) 2(*A
obviously
((-S)d(-S)
=
-
It -
*d(-S))
trivially
is
=
L. The unique characterization -S. properties implies 9 =
invariant
*Q). Finally,
the
under
Q of (-S)
A, of the
mean
curvature
A is
10.3
Surfaces
Willmore
We now turn
in
to the
S3 and Minimal Bdcklund
1-step
Surfaces
transform
d(F+F*)=2*A+2*A* Because S*
We now F is
a
=
use
S,
we can
affine
Bdcklund
initial
F+F*
-S.
with
[,I.
transform
f,
of
L
9
g +
(9.9).
1
Likewise,
for
(10.4)
sphere Sg.
0 1
0
left
From the
0 1
9 -fi
1
g of
entry
properties
of
(10.5)
From Lemma 14
1
then
imply
we
obtain
0)
0
A,
F such that
Then the lower
NO
H
*
(10.4)
(0 1) ( R) (1-f)=(lf (R (1 (1 f) (1 Hg) ( -j ) (1 -Hg) (1 (1 f) (9 HgfI (1 -f) if
2
Hg=f-f,
-R..
=
=
65
H.
=
Nq=-R,
(9.3),
conditions
and
We want to compute the mean curvature Bdcklund transforms we know
see
of L. If dF
I
(7.9)
and
4-Space
Hyperbolic
2*A-2*Q=-dS.
=
(10.3)
choose suitable
coordinates
in
-
ft
-f
0
0
0
1
0
-f 1
-
01
This
implies
H
and -N
-Rg In
particular
on
that
set.
f It
E Im
-
follows
Sg and, because, R
=
=
g
=
H,
N=
=
since
HH,
-
-N +
H
0
=
whence
(f
on
1
-
an
f)H. would
open set
that
(I 1) (
-R
g
0
f
-I
-g)
0 -N +
N and H E R for
f
(f
-
I)H)
O
M-4 Im H =
:
1
R3,
(I H) (-N ; (f-f)H g), (Ig-H) (N+(I-f)H 0) (1H-g) (I g) ( 1-f N+(I-f)H ) (i -g) g
S*
0
-
01
-
0
1
0
1
f
-S9-
f
f
N
01
H
0
0
N
1
0
0
1
-
1
mean w
0
Surfaces
10 Willmore
66
We have
the
shown that
now
thogonally,
and therefore
coordinates
and
are
Euclidean
a
S3
in
mean
g and have the same mean curvature under conformal changes of the ambient
g has
mean
and the
"Micklund
generate
minimal
1w, 2
of
dg
in
hyperbolic
=
but
1 =
4
(SdS
and therefore
+
Let L be
S*
with
i.e.
1
*dS)*
4
This
hyperbolic
in the
If A
is minimal.
remains
property =-
0, then
which may be considered in the (isolated) be singular
g will
w
=
as a
0,
de-
zeros
-S.
(dSS
holomorphic
immersed
an =
I
*dS)
-
minimal
curve,
Then
4
(SdS
+
*dS)
-A,
also
(d From
g.
Therefore,
elsewhere.
converse:
4-space,
as
metric.
constant,
general
In
minimal
Weshow the
A*
'ansform"
surface.
is
or-
that, using affine spheres are tangent
curvature
vector
0, and hence
curvature tr
S'
of g intersect
Weknow
planes.
to
metric,
spheres
curvature
mean
hyperbolic metric, the
Proposition
15
we
*
A)*
=
-d
A.
*
have
(f
I
A
d
4
dw
dw
-f
dw
-dw
f)
f
Therefore
jw-,
dw
f
dw
=
dwf,
and hence
dw(f f
But sition
is not in
S1,
dw=O,
and therefore
*A
=
-A
the backward
conditions To
f)
=
i.e
0. L is Willmore.
Similarly,
Propo-
yields
12
and A*
+
implies Bdcklund
w
=
-77D.
transform
=
(W
Rom S*
h with
dh
-S
1w
we -
know TI
=
-H,
dH and suitable
and
initial
is in Im H = R.
summarize
hermitian > be an indefinite product [11]). Let < lines form an S' C HP', while the two complemen-. isotropic Let L be a Willmore metrics. complete hyperbolic surface tary discs inherit Then a suitable in S' C HP'. forward Bdcklund transform of L is hyperbolic minimal. that is hyperbolic an immersed holomorphic curve Conversely, minimal is Willmore, and a suitable backward Bdcklund transformation is a Willmore surface in S'. (In both cases the Bdcklund transforms may have
Theorem 9 on
IV.
(Richter
Then the
singularities.)
.,.
Spherical
11
In
this
chapter
[8],
Montiel
spheres
suitable
we
which
projection
a
an
following result
in
HP'
affine
[2],
[8]).
Montiel
differs
from
and therefore
what
we
global
[1]
Bryant
A Willmore
have treated
methods
Complex Line Bundles:
of
Degree
of
theorem of
of a holomorphic or anti-holomorphic coordinates, corresponds to a minimal
requires from complex function theory.
11.1
of the earlier
proof
generalizes
(Ejiri
The material
global,
sketch
Surfaces
S3. See also Musso [9].
in
Theorem 10 twistor
Willmore
sphere
so
proof.
and
far:
in
and
Willmore
EEP1 is
Cp3'
in
a
or,
in
The theorem
is
curve
surface
[2]
Ejiri for
in
These
R1.
are
imported
Holomorphicity
complex vector bun le. We keep the symbol J E End(H) for the with the imaginary unit i. endomorphism given by multiplication We denote by R the bundle where J is replaced > is a by -J. If < hermitian metric on E, then Let
E be
a
.,.
R -+ E*
=
E-1, 0
-+
isomorphism of complex vector bundles. Also note that for complex line E1, E2 the bundle Hom(Ei, E2) is again a complex line bundle. There is a powerful for complex line bundles E over a integer invariant Riemann surface: the It classifies these bundles up to isocompact degree. Here two definitions for are the morphism. equivalent degree. is
an
bundles
-
Choose
a
hermitian
Then < R(X,
R(X, Y)
metric
and
a
compatible
Y) 0, 0 > 0 for the curvature tensor (X, Y) J with a real 2-form w E fl2 (M). =
deg(E)
1 :=
27r
F. E. Burstall et al.: LNM 1772, pp. 67 - 72, 2002 © Springer-Verlag Berlin Heidelberg 2002
fm
W.
connection
R of V. Define
V
on
E.
Therefore
68
-
Spherical
11
Choose
Willmore
0
section
a
E V
Surfaces
(E)
with
deg(E)
isolated
0
ord
:=
HP1
in
Then
zeros.
E
:=
indp 0.
O(P)=O of
The index
is defined
0
p of
local
section non-vanishing b z (0) holomorphic parameter z '0 (z) A (z) p. Then 0 (z) for some complex function A : C C U -+ C with isolated zero at 0, and
and
a zero
using
a
where -y is
circle
small
a
fundamental
We state
deg(B)
0.
of the
properties =
=
dA
2-7ri
around
=
ZY A(z)'
1
indp
degree.
deg E-1
=
deg Hom(Ej, E2) More
a
for Mwith
We have
deg E,
-
deg El
+
deg E2.
generally,
deg(El
0
E2)
deg El
deg E2.
+
Example 21. Let M be a compact Riemann surface of genus g, and E its tangent bundle, viewed as a complex line bundle. We compute its degree The curvature of a surface with Riemannian tensor using the first definition. metric < > is given by R(X,Y) K(< Y,. > X- < X,. > Y), where =
.,.
Gaussian
K is the
compatible
with
W(X' Y)
curvature.
=
1traceR 2
(< Y,
_
< Z >
JZ Z
>)
>)
>
Y).
Gauss-Bonnet,
canonical
-
and find
21rX(M)
=
27r(2
-
bundle
Hom(TM, C) ='fw
E
HomR(TM, C) I w(JX)
find
deg(K)
=
2g
-
2.
=
iw(X)}
2g)
.,
>
Complex
11.1
Definition
for
Let E be
11.
a
complex linear complex anti-linear
E is
valued
complex
map
a
Bundles:
Line
bundle.
A
vector
0 from the RE
Holomorphicity
holomorphic of E
sections
69
structure into
the E-
-+.V(KE)
r(E)
:
and
map
a
1-forms a
Degree
satisfying +
6A
Here
if (90
=
:=
!(dA+i*dA). 2
0.
We denote
(Local)
E is
If
vector
then
holomorphic deg E < 0,
maps preserve
then
any
proof and holornorphicity
are called holomorphic, sections of holomorphic
space
and of
In
orientation.
and
structure, index
positive
particular,
V)
E
because
if
M is compact in E vanishes identically.
and
concepts.
If (L, J)
12.
is
a
&
complex line
:=
10
E L
bundle,
then
I JO =,Oil
We define
bundle.
If LI, L2
line
complex quaternionic
deg Lemma15.
holomorphic
isolated
are
global holomorphic section Ejiri theorem we shall apply the concepts of degree to several complex bundles obtained from quaternionic
these
Werelate
Definition
0
of
zeros
of the
In the
ones.
the
with
bundle
line
complex
a
HO(E)\10},
a
the
U.
over
is
0 EF(Eju)
sections
by HO(Eju)
deg EL.
L
line
complex quaternionic
are
bundles,
and Ej
:=
ELj,
then
Homc(El,
Hom+(L1, L2)
is
an
of complex
isomor Phism
vector
bundles.
deg Hom+(L1, L2) The
proof
Example
is
22.
straightforward. Weconsider
immersed
In
deg L,
We now discuss an
E2)
BjEj
B
particular +
one
holomorphic
deg L2 example
in
detail.
curve
LCH=MxEV in
HPI with
vector
bundle,
S. For B E r
mean
the
(K
curvature
complex
End-
(H))
sphere S. The bundle K End- (H) is a complex with being given by post-composition
structure we
define
11
70
Spherical
Surfaces
Willmore
(,9xB)(Y)V5
c9x(B(Y),O)
=
HP1
in
B(OxY)o
-
-
B(Y)o9x0,
where
Oxy:=
0,0 Direct
that
=
I(d+S*d),O,
on on
V, Y1
YD'
J1jx'
+
i90=_I(d-S*d)0f6r'0E.V(H). 2
shows that
computation induced
and 0
2
2
is in fact
this
holomorphic
a
namely
structure,
on
K End-
by 6
-
(H)
K Hom+(TI,
=
H)
TM, and the above (quaternionic) ft.
=
K Homc(ft,
holomorphic
H) 0
structures
on
H
Lemma 16.
(d Proof.
Let
mark
X be
12, and- 0
(d
*
a
E
*
=
holomorphic
local
F(H).
A) (X, JX)o
A) (X, JX)
-
2 (Ox
A) (X).
field,
vector
[X, JX]
i.e.
=
0,
see
Then
(-X
=
A(X)
-
(JX)
-
-
SA(X)
-
A([X, JX])o 1--le-I =0
-(d(A(X),O) + A (X)
(do
+
+
do (X)
*d(SA(X)O))(X)
+ SA(X)
*d(So))
(X)
+
*
d b (X)
A(X) (do
-
S
*
do) (X).
Now
do
+
*d(SO)
=
=
=
=
=
(c9
0
+
Q)o
+ A+
+
*(,9
0
+
+ A+
Q)So
(0+O+A+Q)o+(S0-S5+SA-SQ)So (,9+O+A+Q)o+(-c9+O+A-Q)o 2(6 + A)o 20(A(X)0)
+
2AA(X)O.
Similarly
(o9+5+A+Q)0-S*(o9+O+A+Q)0
do-S*do= =
(a
+
0
+ A+
=
(0
+
0
+ A+
=
2(0
+
A)O.
Q)o Q),0
-
-
S(SO (-,9
SO + SA
-
+
6
-
A+
-
Q)o
SQ)0
Re-
Surfaces
Willmore
Spherical
11.2
71
Therefore
(d
A) (X, JX),o
*
-20x (A(X)O)
=
-2(0x(A(X)0) _2(6xA)(X)06A
=
A E Ho (K EndAs
a
consequence,
exists
and there zeros
bundle
line
a
of A. For local
(H))
0
E
F(L)
6A (Y)o
L
A
0,
_=
or
O(A(Y)O)
=
under
A E Ho (K
Spherical to the
We turn
Proof
(of
A
0.
This
of A
are
=
implies
the
zeros
isolated,
=
the
A(Y)&O.
-
49, like L is structure holomorphic above, get A defines and a K Hom+(TIlL, holomorphic L)
11.2
*
ker A away from C H such that Y we have and holomorphic E HI(TM)
a
we
2A(X)2,0
=0
invariant
is
mark 6. As
d
L
L
=0
Therefore
+
Ho (K Hom+(TI, H)).
=
Lemma23, either
see
2A(X)L9xO
and therefore
L is
that
assume
+
A(X)c9xO)
-
Willmore, 0, and A is holomorphic:
Now
2A(X)2,0
-
Hom+(RIL,
under
invariant on
the
see
Re-
bundle:
of this
section
0,
complex line bundle
L)).
Surfaces
Willmore
.
Theorem
10).
If A
0
=-
or
Q E 0,
then
L is
a
by
projection
twistor
Theorem 5.
Otherwise coincides
we
with
Proposition holomorphic
image of Q almost
the
We have
20.
line
A E
bundle
have the line
We proved
holomorphic
following
the
L)),
JLEHO(KHom+(L,H/L)), if AQ the
=
a
line
bundle
sections
about
in the
then
Q E Ho (K Hom+(HIL,
L
that
of complex
yields
LA
AQEHO(K Hom+(HIL, L)) 2
J1
0 then
statement
appendix. The degree formula
others
similarly everywhere. and
bundles:
Ho (K Hom+
and
L,
E
Ho (K Hom+(L, HIL))
A. We give
the
(similar)
proofs
of the
72
Spherical
11
Willmore
ord
Surfaces
JL
ord(AQ)
=
deg
=
2
deg
K
3
deg
K
6(g
S2,
For M=
i.e.
g
=
0,
K
-
IFffP1
in
deg
-
-
+
deg
L
6L
ord
-
get ord(AQ)
we
deg HIL
deg HIL
-
1)
L +
ordJL.
