Francis E. Burstall, Dirk Ferus, Katrin Leschke, Franz Pedit, Ulrich Pinkall - Conformal Geometry of Surfaces in Quaternions-Springer (2002)

3
Berlin
Heidelberg
New York
Barcelona
Hong Kong
London
Milan
Paris
Tokyo
RE. Burstall D. Ferus
K. Leschke R Pedit U. Pinkall
Conformal Geometry
of Surfaces in S4
and Quaternions
1011. Springer
I.,
Authors
Francis E. Burstall Franz Pedit
Dept. of Mathematical Sciences Dept. of Mathematics

University of Bath and Statistics
Claverton Down University of Massachusetts
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 Ulrich Pinkall
Technical University of Berlin Technical University of Berlin
MA 8-3 MA 8-3
Strasse des 17. Juni 136 Strasse des 17. Juni 136
10623 Berlin, Germany 10623 Berlin, Germany
E-mail. ferus@math. tu-berlin.de E-mail. pinkall@math.tu-berlin.de
E-mail: leschke@math.tu-berlin.de
Cover figure from D. Ferus, R Pedit:Sl-equivariant Minimal Tori in S' and Sl-equivariant Willmore Tori
in S3. Math. Z. 204,269-282 (199o)
CatalogIng-in-Publication Data applied for.

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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 firstcomprehensive introduction to the authors' recent attempts

toward a understanding, of the global concepts behind spinor represen-
better
tations of surfaces in 3-space. The important new aspect is a quaternionic-
valued function theory, whose "meromorphic functions" are conformal maps
into ff- which extends the classical complex function theory on Riemann sur-
faces. 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, holo-
morphic curves in projective space, Kodaira embedding, and Riemann-Roch,
carry over to the quaternionic setting. Additionally, an important new in-
variant of the quaternionic holomorphic theory is the Willmore energy. For
quaternionic holomorphic curves in HP' this energy is the classical Willmore
energy of conformal surfaces.
The present lecture note is based given by Dirk Ferus at the
on a course
Summer School on Differential Geometry at Coimbra in

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 con-
formal 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 immer-
sions 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
R3 and that of Willmore surfaces in EEP1, and use it to construct Darboux

transforms for the latter.
Bath/Berlin, Francis Burstall, Dirk Ferus, Katrin Leschke,

August 2001 Franz Pedit, Ulrich Pinkall
'Table of Contents
I Quaternions ..............................................
1
1.1 The Quaternions .......................................
1
1.2 The Group S3 .......................................... 3
2 Linear Algebra over the Quaternions ..............

5
2.1 LinearMaps, Complex Quaternionic Vector Spaces .........
5
2.2 Conformal Maps .......................................
7
3 Projective Spaces .........................................

9
3.1 Projective Spaces and Affine Coordinates ..................
9
3.2 Metrics on HPI ......................................... 11
3.3 Moebius Transformations on HP . ........................
13
3.4 Two-Spheres in S4 ......................................
13
4 Vector Bundles ...........................................

15
4.1 Quaternionic Vector Bundles .............................
15
4.2 Complex Quaternionic Bundles ...........................
18
4.3 Holomorphic Quaternionic Bundles .......................
20
5 The Mean Curvature Sphere .............................

23
5.1 S-Theory ...............................................
23
5.2 The Mean Curvature Sphere .............................
24
5.3 Hopf Fields ............................................
27
5.4 The Conformal Gauss Map ..............................
29
6 Willmore Surfaces ........................................ 31

6.1 The Energy Functional ..................................
31
6.2 The Willmore Functional ................................
35
7 Metric and Affine Conformal Geometry ..................

39
7.1 Surfaces in Euclidean Space ..............................
39
7.2 The Mean Curvature Sphere in Affine Coordinates ..........
42
7.3 The Willmore Condition in Affine Coordinates .............
44
VIII Table of Contents
8 Twistor Project"ions ...................................... 47

81 Twistor Projections ..................................... 47
8.2 Super-Conformal Immersions ............................. 50
9 Bhcklund Transforms of Willmore Surfaces ...............

53
9.1 Bdcklund Transforms ................................... 53
9.2 Two-Step Bdcklund Transforms ..........................
57
10 Willmore Surfaces in S3 .................................. 61

10.1 Surfaces in S ........................................... 61
10.2 Hyperbolic 2-Planes .................................... 63
10.3 Willmore Surfaces in S3 and Minimal Surfaces in Hyperbolic
4-Space ............................................... 64
11 Spherical Willmore Surfaces in HPI ...................... 67

11.1 Complex Line Bundles: Degree and Holomorphicity .........
67
11.2 Spherical Will1nore Surfaces ............................. 71
12 Darboux tranforms ....................................... 73

12.1 Riccati equations ....................................... 73
12.2 Constant mean curvature surfaces in R3 ...................
74
12.3 Darboux transforms of Willmore surfaces .................. 79
13 Appendix ......
........................................... 83
13.1 The bundle L .......................................... 83
13.2 Holomorphicity and the Ejiri/Montiel theorem .............
84
References ................................... .................

87
Index .......................................................... 89
I Quaternions
1.1 The Quaternions
The Hamiltonian quaternions H are the unitary R-algebra generated by the
symbols i, j, k with the relations
i2 = j2 = k2 =
_1'
ii --
-ii =
k, ik =
-kj =
i) ki = -ik
The multiplication is associative but obviously not commutative, and each

non-zero element has a multiplicative inverse: We have a skew-field, and a
4-dimensional division algebra over the reals. Frobenius showed in 1877 that
R, C and H are in fact the only finite-dimensional R-algebras that are asso-
ciative and have no zero-divisors. For the element
a =
ao + ali + a2i + a3k, al C- R, (1.1)
we define
a:= ao -
ali -
a2i -
a3k,
Rea:= ao,
Ima:= ali + a2i + a3k.
Note that, in contrast with the complex numbers, Im a is not a real number,
and that conjugation obeys
Wb_ = b a.
Weshall identify the real vector space H in the obvious way with R, and
the subspace of purely imaginary quaternions with R3:
W = IMH.
The reals are identified with RI. The embedding of the complex numbers
C is less canonical.quaternions The i,-j,k equally qualify for the complex
imaginary unit, any.purely

and in imaginaryfact quaternion of square -1
would do the job. Rom now on, however, we shall usually use the subfield
C C ffff generated by 1, i.
F. E. Burstall et al.: LNM 1772, pp. 1 - 4, 2002

© Springer-Verlag Berlin Heidelberg 2002
2 1 Quaternions
Occasionally we shall need the Euclidean inner product on R4 which can
be written as
< a, b >R= Re(ab) =

Re(ab) (ab + ba).
2
Wedefine
a a, a >R = vfa- d.
Then
jabj =
jal Ibl. (1.2)
A closer study of the quaternionic multiplication displays nice geometric as-
pects.
Wefirst mention that the quaternion multiplication incorporates both the
usual vector and scalar products on V. In fact, using the representation (1.1)
one finds for a, b E Im Eff = R'
ab=axb- <a,b>R- (1-3)

As a consequence we state
Lemma1. For a, b G H we have
1. ab = ba
if and only if Im a and Im b are linearly dependent over the reals.
In particular, the reals are the only quaternions that commute with all
others.
2. a' = -1 if and only if Jal = 1 and a = Im a. Note that the set of all such
a is the usual two-sphere
S2 C V = IMH.
Proof. Write a =
ao + a', b =
bo + Y, where the prime denotes the imaginary
part. Then
ab =
aobo + aob' + a'bo + a'b'
=
aobo + aob' + a'bo + a' x Y- < a', Y >R -
All these products, except for the cross-product, are commutative, and (1)
follows. Romthe same formula with a = b we obtain Im a 2 2aoa'. This
=
vanishes if and only if a is real or purely imaginary. Together with (1.2) we

obtain (2).
1.2 The Group S3 3
1.2 The Group S'
The set of unit quaternions
S3 := Ip E HI IM12 =
1}
i.e. the 3-sphere in H = RI, forms a group under multiplication. We can also
interpret it as the group of linear maps x -+ px of H preserving the hermitian
inner product
< a, b >:= 51b.
This group is called the symplectic group Sp(l).

We now consider the action of S3 on H given by
S3 X H __ H, (/-t, a) 1-4 pap-1.
By (1.2) this action preserves the norm on H= R' and, hence, the Euclidean
scalarproduct. It obviously stabilizes R C H and, therefore, its orthogonal
complement R' = Im H. We get a map, in fact a representation,
,X : '53 -+ SO(3), y -+ y ... P-1 II.H-
Let us compute the differential of 7r. For p E S3 and v E Tt'S3 we
get
dl -,ir(v)(a)
-
= vaIL-1 -
pap-1vp-1 =
p(p-'va -
ap-'v)/-t-1.
Now y-1v commutes with all a E ImH if and only if v =

rp for some real
r. But then 0, v =
p. because v diffeomorphism
I of S3
Hence ir is a local
onto the 3-dimensional manifold SO(3) of orientation preserving orthogonal
transformations of R3. Since S3 is compact and SO(3) is connected, this is
a covering. And since pap-I a for all a E ImH if and only if p E R, i.e. if
=
and only if M 1, this covering'is

= 2:1. It is obvious that antipodal points of
S3 are mapped onto the same orthogonal transformation, and therefore we
see that
SO(3)- S31111 Rp3
Wehave now displayed the group of unit quaternions as the universal covering
of SO(3). This group is also called the spin group:
S3 = SP(j) =
Spin(3).
If we identify H = C G) C?, we can add yet another isomorphism:
S3 - SU(2).
4 I Quaternions
In fact, let p =
Ito + 1-iij C- S' with po, pi E C. Then for a,
-
C- R we have
j (a + iP) (a = -
i,8)j. Therefore the C-linear map AIL : C2 -4 C2 ,
x 1-4 fix
has the following matrix representation with respect to the basis 1, j of 0:
AA1 =
yo + jLjj =
Imo + ifil
A,.j =-/-tl + poj =1(-pi) + jpo.
Because of yopo + /-tlpj. =

1, we have
( Po
tL1 f4l)
Po
E SU(2).
2 Linear Algebra over the Quaternions
2.1 Linear Maps, Complex Quaternionic Vector Spaces
Since we consider vector spaces V over the skew-field of quaternions, there are
two options for the multiplication by scalars. We choose quaternion vector
spaces to be right vector spaces, i.e. vectors are multiplied by quaternions

from the right:
V x H -+ V, (v, A) F-+ vA.
The notions dimension, subspace, and linear map work as in the usual
of basis,
commutative algebra. The same is true for the matrix representation
linear
of linear maps in finite dimensions. However, there is no reasonable definition
for the elementary symmetric functions like trace and determinant: The linear
map A : H -+ IH x -+ ix, has matrix (i) when using 1 as basis for H, but
matrix (-i) when using the basis j.
If A E End(V) is an endomorphism, v E V, and \ E H such that
Av =
vA,
then for any p E H\ f 01 we find
A(vp) =
(Av)p =
vAp =
(vp)(p-lAp).
If A is real then the
eigenspace is a quaternionic subspace. Otherwise it is
a real -
but quaternionic
not a vector subspace, and we obtain a whole
-
2-sphere of "associated eigenvalues" (see Section 1.2). This is related to the

fact that multiplication by a quaternion (necessarily from the right) is not
an H-linear endomorphism of V. In fact, the space of Iff-linear maps between
quaternionic vector spaces is not a quaternionic vector space itse f.
Any quaternionic vector space V is of course a complex vector space,
but this structure depends on choosing an imaginary unit, as mentioned in
section 1.1. We shall instead (quite regularly) have an additional Complex
structure on V, acting from the left, and hence commuting with the quater-
nionic structure. In other words, we consider a fixed J E End(V) such that
J' = -1. Then
(X + iy)v := VX + (Jv)y.

6 2 Linear Algebra over the Quaternions
In this case we call (V, J) a complex quaternionic (bi-)vector space. If. (V, J)
and (W, J) such spaces, then the quaternionic
are maps from V to W
linear
split as a direct sum of the real vector spaces of complex linear (AJ = JA)
and anti-linear (AJ =
-JA) homomorphisms.
Hom(V, W) =
Hom+(V, W) ED Hom- (V, W)
In fact, Hom(V, W) and Hom(V, W) are complex vector space with multi-
plication given by
(x + iy) Av : =
(Av) x + (JAv) y.
The standard
example of a quaternionic vector space is H" An example .
of complex quaternionic,
a vector space is HI with J(a, b) := (-b, a).
On V H, any complex structure
= is simply left-multiplication by some
N E IHI with N2 -1. The following
= lemma describes a situation that
naturally produces such an N, and that will become a standard situation for
us. But, before stating that lemma, let us make a simple observation:
Remark 1. On a real 2-dimensional vector space U each complex structure

J E End(U) induces an orientation 0 such that (x, Jx) is positively oriented
for any x .0 0. Wethen call J compatible with 0.
Lemma2 (Fundamental lemma).

1. Let U C H be a real subspace of dimension 2. Then there exist N, R (-= H
with the following, three properties:
2 2
N = -
I = R (2.1)
NU= : U =
UR, (2.2)
U =
fx E HI NxR =
x}. (2-3)
The pair (N, R) is unique up to sign. If U is oriented, there is only one
such pair such that N is compatible with the orientation.

2. If U, N and R are as above, and U C Im H, then
N= R,
and this is a Euclidean unit normal vector of U in Im H = R3.

3. Given N, R E H with 1V' = -1 = R2, the sets
U:=jxEHjNxR=x}, Ul :=IxEHINxR=-x}
are orthogonal real subspaces of dimension 2.
Definition 1. Motivated by (2) of the lemma, N and R are called a left and
right normal vector of U, though in general they are not at all orthogonal to
U in the geometric sense.
2.2 Conformal Maps 7
Proof (of the lemma). (1). If I E U and if a E U is a unit vector orthogonal

to 1, then a2 = -
1. Hence (N, R) =
(a, -a) works for U, and the uniqueness,
up to sign, follows easily from NI E U and Na E U. If U is arbitrary, and
x E U\10} then put U := x-'U. Clearly, 1 E U. Moreover, (N, R) works for
U if and Nx, R)
only if (x
-
1
works for 0.
(2). If R, and u, v is an orthonormal
U C Im H = basis of U, then
N = R u x v = uv satifies the requirements:
= Use the geometric properties
of the cross product.
(3). The above argument shows that u(x) := NxR has I-eigenspaces of
real dimension 2. Since a is orthogonal, so are its eigenspaces.
Example 1. Let (V, J), (W, J) be complex quaternionic vector spaces of di-
mension 1. Then Hom+(V, W) is of real dimension 2. To see this, choose bases
v and w, and assume
Jv =
vR, Jw = wN.
Then N2= -1 =7R'. Now F E Hom(V, W) is given by F(v) =

wa, and
FJ = JF -#= FJv = JFv
4==> waR = J(wa) =

(Jw)a = wNa 4=* aR = Na.
But the set of all such a is of real dimension 2, by the last part of the lemma.
The same result holds for Hom- (V, W). As stated earlier, Hom(V, W) are
complex vector spaces, and therefore (non-canonically) isomorphic with C.
2.2 Conformal Maps
A linear map F : V -+ Wbetween Euclidean vector spaces is called conformal

if there exists a positive A such that
< Fx,Fy >= A < x,y >
for all x, y E V. This is equivalent to the fact that F maps a normalized

orthogonal basis of V into a normalized orthogonal basis of F(V) C W. Here
"normalized" all vectors have the same length,
means that possibly 54 1.
If V = W= R'
C, and J : C -+ C denotes multiplication
=
by the
imaginary unit, then J is orthogonal. For x E C, IxI 4- 0, the vectors (x, Jx)
form a normalized orthogonal basis. The map F is conformal if and only if
(Fx, FJx) is again normalized orthogonal. On the other hand (Fx, JFx) is
normalized orthogonal. Hence F is conformal, if and only if
FJ =
JF,
where the sign depends on the orientation behaviour of F.

2 Linear Algebra over the Quaternions
Note that this condition does not involve the scalar product, but only
involves the complex structure J. A generalization of this fact to quaternions
is fundamental for the theory presented here.
If F : R' = C -+ R4 = H is R-linear and injective, then U =
F(W) is a
real 2-dimensional subspace

by J. Let N, R E H be its left and of H, oriented
right normal vectors. Then NU U UR, and N induces an orthogonal = =
endomorphism of U compatible with the Euclidean scalar product of V. The

map F : R' -+ U is conformal if and only if FJ = NF. Hence F : C -+ H is
conformal if and only if there exist N, R E H, NI = -I =
RI, such that
*F:= FJ = NF = -FR.
This leads to the following fundamental
Definition 2'. Let M be a Riemann surface, i.e. a 2-dimensional manifold

endowed with complex structure
a J: TM-+ TM, j2 -1. = A map f : M
H R4 is called conformal, if there exist N, R : M -+ H such that with
*df := df o J,
N2= -1
2
= R (2.4)
*df =
Ndf =
-dfR. (2.5)
If f is an immersion then (2.4) follows from (2.5), and N and R are unique,
called the left and right normal vector Of f.
Remark 2. -
Equation (2.5) is an analog of
*df =
idf
for functions f : C -+ C, i.e. of the Cauchy-Riemann equations. In this

sense conformal maps into H are a generalization of complex holomorphic
maps.
-
If f is an immersion, then df (TpM) C H is a 2-dimensional real subspace.
Hence, according to Lemma 2, there exist N, R, inducing a complex struc-
ture J on TM1--- df (TpM). The definition requires that J coincides with
the complex structure already given on TpM.
-
For an immersion f the existence of N : M -4 H such that *df Ndf =
already implies that the immersion f : M-+ H is conformal. Similarly for

R.
-
If f : M-+ Im H RI is an immersion then N = R is "the classical" unit
normal vector of But for general f : M --+ H, the vectors N and R are
not orthogonal to df (TM).

3 Projective Spaces
In complex function theory the Riemann sphere CP1 is more convenient as

a target space for holomorphic functions than the complex plane. Similarly,
the natural target space for conformal immersions is HP1, rather than H. We
therefore give a description of the quaternionic projective space.
3.1 Projective Spaces and Affine Coordinates
The quaternionic projective space HP' is defined, similar to its real and
complex cousins, as the set of quaternionic lines in H1+1. We have the (con-
tinuous) canonical projection
7r : Ep+1 \f 01 __+ ff-lpn, X + 7r(X) =

[X] = XH.
The manifold structure of Rpn is defined as follows:

For any linear form P E (Hn+') P 54 0,
U: 7r(x) -+ X < P, X >_1
is well-defined and maps the open set 17r (x) I < P, x > 54 0} onto the affine
hyperplane P =
1, which is isomorphic to Hn. Coordinates of this type are
called affine coordinates for Hpn. They define a (real-analytic) atlas for ffffpn.
We shall often use this in the following setting: We choose a basis for
H1+' such that # is the last coordinate function. Then we get
XJX-1
(Xnx*n+l)
X1 n+
XlXn+1
or
Xn Xnxn+l i I
Lxn+lj
The set
17r(x) I < 0, X >= 01

is called the hyperplane at infinity.

10 3 Projective Spaces
Example 2. In the special case n =

1, the hyperplane at infinity is a single
point: HP' is the one-point compactification of R4, hence "the" 4-sphere:
Elp,
Note however, that the notion of the antipodal map is natural on the usual
4-sphere, but not on HP' -
unless we introduce additional structure, like a
metric.
For our purposes it is important to have a good description of the tangent

space Tilapn for 1 E ffffpn. For that purpose, we consider the projection
7: ffp+l \101 _-, Hpn
in affine coordinates: If # E (Hn+')* is as above, then
h = u o 7r : IT+' \10} ffP+l, x -+ x < 0, x
satisfies
d h(v)=v<#,x>-'-x<P,x>-'<P,v><O,x>- .
Therefore
ker dx h =
xH,
dx,xh(vA) =
dxh(v)
for A E H\10}, and the same holds for 7r:
ker d,, 7r xH, (3-1)

dx,\7r(vA) dx7r(v). (3.2)
By (3.1), dx7r induces an isomorphism

'-
dx7r : EP+1 /1 . Tllffpn, 1
of real vector spaces, but it, depends on the choice of x E 1. To eliminate this
dependence, we note that by (3.2) the map'
Hom(l, ffP+1 /1) _+ T1 Rffpn, F -+ dx -7r (F (X)),
with x E 1\101 is a well-defined isomorphism:,
Hom(l, RP+l /1) c-

_
T Hpn. (3.3)
In other words, we identify dx ir (v) with the homomorphism from 1 = 7r (x) =
xH to Hn+'/l that maps x to 7r, (v) := v mod 1. For practical use, we rephrase
this as follows:
3.2 Metrics on HP' 11
Proposition .1. Let 1: M-4 H'+1 \10} and f = 7rl: M_+ Elpn.
Let p E M, 1:= f (p), v, E TpM. Then
dpf : TpM ---> Tf (p)Hpn =

Hom(f (p), ffV+1 If (p))
is given by
dp f (v) (f(p) A) =
7rl (dp f (v) A).
We denote the differential in this interpretation by Jf:
16f (v) (f) =

df (v) (3.4)
Proof. The tangent vector
dp f (v) =
di(p) ir (dp 1(v)) E Tf (p)
Hpn
is identified with the homomorphism F : f (p) --+ Hn If (p), that, maps j(p)
into dpl(v) mod f (p).
3.2 Metrics on Hpn
Given a non-degenerate quaternionic hermitian inner product < .,.

> on
Hn+l, we define a (possibly degenerate Pseudo-) Riemannian metric on Hpn

as follows: For x E Hn+1 with < x, x > 54 0 and v, w E (x4 we define
1
< d, 7r (v), dx 7r (w) > =- Re < v,w >
< X'x >
This is well-defined since, for 0 54 A E H, we have
< d ,.Xir(vA), dxX7r(wA) >=< d ,7r(v), dx7r(w) >
It extends to arbitrary v, w by
< dx 7r (v), dx 7r (w) > = Re<V, WX< x, W> < x, x > >
< X'X >

< V'W >< X'X > < V,X >< >
IX'W
-
Re (3-5)
< X'X >2 .
Example 3. For < v, w >= 1: ITkWk we obtain the standard Riemannian met-
ric on IRpn. (In the complex case, this is the so-called Fubini-Study metric.)
The corresponding conformal structure is in the background of all of the
following considerations.
Wetake this standard Riemannian metric on DIP' = S' and ask which metric
it induces on R4 via the affine parameter
h : H -+ HP',x t-+
1XI _
Let h : H -+ H2, x -+ (x, 1), and let "=-" denote equality mod
(X)
1
ff.
Then
6xh(v)((x)) (v) (v0) (x) VV)

'7-
xv I
1
d,,h(v) 0
=_ -
1 1 + xx -
-
-,t I + xx-,
The latter vector is < .,. >-orthogonal to (x, 1), and therefore the induced
metric on H is given by
V) )
1 V W
h* < v,w >x = Re < >
(1 + X.:t)3 -.t W
Re(;vw) < V,W >R

(I + X;,-) 2
(1 + Xj )2
But stereographic projection of S4 induces the metric
4
< V5 W >R
(I+ Xj )2
on R4. Hence the standard metric on HP' is of constant curvature 4.
Example 4. If we consider an indefinite hermitian metric on Hn+', then the

above construction of a metric on Hpn fails for isotropic lines (< 1, 1 >= 0),
but these points are scarce. We consider the case n =
1, and the hermitian
inner product
< V, W >= ITIW2 + IT2W1
Isotropic lines are characterized in affine coordinates h : x -+

(xl) by
(X) (X)
,
0 =< >= j + X,
I ,
I
i.e. by x E ImH = R3.
The point at infinity (1)

0
H is isotropic, too. Therefore, the set of isotropic
points is a 3-sphere S3 C S', and its complement consists of two open discs
or -
in affine coordinates -
two open half-spaces.
As in the previous example, we find
h* < v,w Re ( v- < ViW >R

(2 Re X)2 2 ReX)2
for the induced metric on the half-spaces Re :A 0 of H. This is -
up to a
constant factor -
the standard hyperbolic metric on these half-spaces.
3.4 Two-Spheres in S4 13
3.3 Moebius Transformations on HP1
The group Gl(2, 4 acts on RP' by G(vH) := GvH. The kernel of this action,
i.e. the set of all GE G1(2, P such that Gv E vH for all v, is jjol I p E Rj.
How is this action compatible with the metric induced by a positive defi-
nite hermitian metric of V? Using (3-5) we find
" G(O), G(vA) >< Gx, Gx > < G(vA), Gx > < Gx, G(vA) >
IdG(d,,-x(vA))I'
-
= Re
Gx,Gx >2 <
< Gv, Gv >< Gx, Gx > < Gv, Gx >< Gx, Gv >
IA12
-
= Re
< Gx, Gx >2
=
1,\12 IdG(d,,7r(v)) 12
Taking G = I we see that for v 54 0 the map
H -+ T,( ,)EIP', A -+ d,,7r(vA)

is length-preserving up to a constant factor, isomorphism.
i.e. is a conformal
But the same is obviously true for the metric induced
by the pull-back under
an arbitrary G, and therefore GL(2, R acts conformally on RP
1
S4. We =
call these transformations the Moebius transformations on Rp'. In affine

coordinates they are given by
(a db) [x] [ax db11 [(ax

c
=
cX
+
+
=
+ b) (cx
I
+ d)
This emphasises the analogy with the complex case.
It is known that this is the full group of all orientation preserving confor-
mal diffeomorphisms of S4, see [7].
3.4 Two-Spheres in S4
We consider the set
Z =
IS E End(EO) I S2
For S E Z
-
we define
S' := fl E RP1 I Sl =
11. ,
We want to show
Proposition 2. 1. S' is a 2-sphere in KPI, i.e corresponds to a real 2-

plane in H R4 under a suitable =
affine coordinate.
2. Each 2-sphere can be obtained in this way by an S E Z, unique up to
sign. -
Proof Weconsider H' as a (right) complex vector space with imaginary unit
i. Then S is Clinear and has a (complex) eigenvalue N. If Sv vN, then
=
S(vH) = vN H = vH.
Hence S' 0.
We choose a basis v, w of EV such that vH E S', i.e. Sv vN for some
N, and Sw = -vH -
wR. Then S' = -1 implies
N2 = -1 = R2, NH= HR.
For the affine parametrization h : H -+ IMP', x [vx + w] we get:
[vx +,w] E S' 3 S(vx + w) =

(vx + w)-y
3.y vNx -
vH -
wR =
vx-y + w7
Nx -
H = x-y
37
-R -y
Nx + xR H.
This is a real-linear equation for x, with associated homogeneuos equation
Nx + xR = 0.
By Lemma2 this is of real dimension 2, and any real 2-plane can be realized
this way.
Obviously, S and -S define the same 2-sphere. But S determines (N, R),
thus fixing an orientation of the above real 2-plane and thereby of S'. Hence
the lemma can be paraphrased as follows:
Z is the set of oriented 2-spheres in S4 = fflpl.

4 Vector Bundles
We shall need vector bundles over the quaternions, and therefore briefly in-
troduce them.
4.1 Quaternionic Vector Bundles
A quaternionic vector bundle 7r : V -+ Mof rank n over a smooth manifold

Mis a real vector bundle of rank 4n together with a smooth fibre-preserving
action of H on V from the right such that the fibres become quaternionic
vector spaces.
Example 5. The product bundle 7r : M x Hn -+ M with the projection on
the first factor and the obvious vector space structure on each fibre f x} x Hn
is also called the trivial bundle.
Example 6. The points of the projective space Elpn are the 1-dimensional
subspaces of Hn+'. The tautological bundle
7r_, : Z ___, Hpn
is the line bundle with Zi = 1. More precisely
_p := 1 (1, V) E lffpn X Ep+1 I V C 1},

7rz Rpn, (1, V) _+ 1.
The differentiable and vector space structure are the obvious ones.
Example 7. If V -+ Mis a quaternionic vector bundle over M, and f : M

1 1 is a map, then the "pull-back" f *V 4Mis defined by
f*V:= I(x,v) IV E Vf(,,)} CMX V
with the obvious projection and vector bundle structure. The fibre over x E M
is just the fibre of V over f (x).

16 4 Vector Bundles
We shall be concerned with maps f : M -4, RP' from a surface into the
projective space. To f we associate the bundle L : =
f Z,
*
whose fibre over
x is f (x) c ffP+l =
JxJ x H1+1. The bundle L is a line subbundle of the
product bundle
H:= Mx EP+l.
Conversely, every line subbundle L of H over Mdetermines a map f : M

HP' by f (x) := L,,. We obtain an identification
Maps Line subbundles

f : M_+ ffffpn L c H = Mx Hn+1
All natural constructions for vector spaces extend, fibre-wise, to opera-

tions in the category of vector bundles. For example, a subbundle L of a
vector bundle H induces a quotient bundle HIL with fibres H,;IL_-. Given
two quaternionic vector bundles V1, V2 the real vector bundle Hom(V1, V2)
has the fibres Hom(Vi., V2 ). A section !P E F(Hom(Vi, V2)) is called a vec-
tor bundle homomorphism. It is a smooth map !P : V, -- V2 such that for

all x the restriction 4ilv,. maps V1,; homomorphically into V2x. There is an
obvious notion of isomorphism for vector bundles.
Example 8. Over lffpn we have the product bundle H = HP' x H1+1 and,
inside it, the tautological subbundle Z. Then
THP' Hom(Z, HIZ),
see (3.3).
Example (and Definition).

9 Let L be a line subbundle of H = Mx Hn+'.
Let 7rL -+ HIL :E -P(Hom(H, HIL))
H be the projection. A section 0 c
.P(L) C F(H) is a particular map M -+ Hn+1 .

If X E TpM, then
dO(X) E Hp Hn+', and =
WL (do (X)) E (HIL)p = EP+1 ILp.
Let A: M-+ R Then
7rL (d(0A) (X)) =

7rL (dO(X)I\ + Odl\(X)) :--:
7rL (dO(X))I\-
Wesee that
0 -+ 7rL (4 (X)) :;--: 6 (X) (0)
is tensorial in 0, i.e. we obtain
J(X) :-- JL (X) G Hom(Lp, (HIL)p).

4.1 Quaternionic Vector Bundles 17
Of course this is R-linear in X as well, so 6 should be viewed as a 1-form on
Mwith values in Hom(L, HIL):
IJ E f2l (Hom(L, HI (4.1)
Let us repeat: Given p E M, X E TpM, and 00 E Lp, there is a section
0 E F(L) such that 0(p) =

Oo. Then
I Jp(X),Oo =
7rL(dv0(X)) =
dpV)(X) m
Note the similarity to the second fundamental form
1
a(X, Y) =
(dY (X)) .
of a submanifold Min Euclidan space. In the case at hand, L corresponds

to TM and H1+1 A correspond 's to the normal bundle. This is the general
method to measure the change of a subbundle L in a (covariantly connected)
vector bundle H.
We can view L as a map f : M -+ HP'. Even if this is an immersion,
6 clearly has nothing to do with the second fundamental form of f. Instead,
comparison with Proposition I shows that
6: TM-+ Hom(L, HIL)
corresponds to the derivative of f ,

and we shall therefore call it the derivative
of L .
Example 10. The dual V` := fw : V -4 Hjw H-linearl of a quaternionic

vector space V is, in a natural way, left a H-vector space. But since we choose
quaternionic vector spaces to be right vector spaces, we use the opposite
structure: For w E V* and A E H we define
wA := w.
This extends to quaternionic vector bundles. E.g., if L is a line bundle, i.e. of

rank 1, then L* is another quaterionic line bundle, usually denoted by'L-1.
A quaternionic vector bundle isomorphic is called

with the trivial if it is
product bundle Mx H, global
i.e. if there 0,, : M-+ V exist sections ,
that form a basis of the fibre everywhere. Note that for a quaterniQnic line
bundle over a surface the total space V has real dimension 2+4 6, and hence =
any section 0 : M -+ V has codimension 4. It follows from transversality

theory that any section can be slightly deformed so that it will not hit the
0-section. Therefore there exists a global nowhere vanishing section: Any
quaternionic line bundle over a Riemann surface is (topologically) trivial.
18 4 Vector Bundles
4.2 Complex Quaternionic Bundles
A complex quaternionic vector bundle is a pair (V, J) consisting of a quater-

nionic vector bundle V and a section J E 1'(End(V)) with
j2
see section 2.1.
Example 11. Given f : M -+ H, *df Ndf the quaternionic =

,
line bundle
L = Mx H has a complex structure given by
Jv:= Nv.
Example 12. For a given S E End(H2) with S2 =

-1, we identified
S' =
111 Sl =
1} C HP,
as a in RP1, see. section

2-sphere 3.4. Wenow compute J, or rather the image
of 6, for the
corresponding line bundle L. In other words, we compute the
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
6SO =
7rLd(SO) =
7rLSdV) =
S7rLdo =
S60.
This shows
TS' =
image 6 C Hom+(L, HIL).
But the real vector bundle Hom+(L, HIL) has rank 2, see Example 1, and
since S' is an embedded surface, the inclusion is an equality:
TIS' =
Hom+(LI, (HIL)I) C Hom(LI, (HIL)j) = TjHP1.
For our next example we generalize Lemma2.
Lemma3. Let V, W'be 1-dimensional quaternionic vector spaces, and
UC Hom(V, W)
be a 2-dimensional real vector subspace. Then there exists a pair of complex
structures J E End(V), j E End(W), unique up to sign, such that
ju = U= Ui,
U =
fF E Hom(V, W) I jFJ =
-Fj
If U is oriented, then there is only one such pair such that J is compatible
with the orientation.
Note: Here we choose the sign of J in such a way that it"corresponds to
-R rather than R.
4.2 Complex Quaternionic Bundles 19
Proof. Choose non-zero basis vectors v E V, w E W. Then elements in
Hom(V, W) and endomorphisms of V or of Ware represented by quaternionic

1 x 1-matrices, and therefore the assertion reduces to that of Lemma2.
The following is now evident:
Proposition 3. Let L C H = M x W be an immersed oriented surface in

EIPI with derivative 5 E J?I (Hom(L, HIL)). Then there exist unique complex
structures on L and HIL, denoted by J, j, such that for all x E M
h(TXM) 6(T,:M) =
6(Txm)j,
J6 =
ji,
and J is compatible with the orientation induced by J: T ,M -+ 6(TxM).

Definition 3. A line subbundle L C H = M x Hn-- ' over a Riemann sur-
face M is called conformal or a holomorphic curve in Rpn, if there exists a
complex structure J on L such that
*6 = 6j.
From the proposition we see: If L is an immersed holomorphic curve in
HPI, i.e. if 6 is in addition injective, such that J(TM) C Hom(L, HIL)

is a real subbundle of rank 2, then there is also a complex structure J E
1'(End(H/L)) such that
*6 = jj (4.2)
A Riemann surface immersed into HP1 is a holomorphic curve if and only

if complex structures
the given by. the proposition are compatible with the
complex structure given on Min the sense of (4.2).
Example 13. Let f : M- H be a conformally immersed Riemann surface

withright normal vector R, and let L be the line bundle corresponding to
: M-* HPI.
Then E.F(L), and
J( (f) I
R) =
7rLd((fl) R) 7rL((dof) R+
(fl) dR)
=
7rL
d
'") 0
=
-7rL
(*Of)d (fl)
If we define J E End(L) by J R then
20 4 Vector Bundles
jj =
*J,
hence (L, J) is a holomorphic curve. Conversely, if (L, J) is a holomorphic
curve,then J
M M 1
= -
I
R for some R : M-+ E and f is conformal with
right normal vector R.
4.3 Holomorphic Quaternionic Bundles
Let (V, J) be a complex quaternionic vector bundle over the Riemann surface
M. We decompose
HomR(TM, V) KV (D kV,
where
KV:= jw: TM V I * w =
Jwj,
KV:= jw: TM-+ V I * w =
-Jwl.
Definition 4. A holomorphic structure on (V, J) is a quaternionic linear

map
D : P (V) -4 P (EV)
such that for all 0 E 1'(V) and A: M-+ H
D(OA) =
(DO)A + 1(OdA + JO
2
* dA). (4.3)
A section 0 E.P(V) is called holomorphic if Do =

0, and we put
HO(V) = ker D C F (V).
Remark 3. 1. For a better understanding of this, note that for complex-

valued A the anti-C4inear part (the k-part) of dA is given by OA
-1 (dA + i * dA). In fact,
(dA + i * dA) (JX) =

*dA(X) -
i dA(X) =
-i(dA + i * dA) (X).
A holomorphic structure is a generalized 0-operator. Equation (4.3) is the

only natural way to make sense of a product rule of the form "D(,OA)
(D,O)A + OOA".
2. If L is holomorphic
a curve in ELI"', does this mean L carries a natural
holomorphic structure? This is not yet clear, but we shall come back to
this question. See also Theorem I below.
4.3 Holomorphic Quaternionic Bundles 21
Example 14. Any given J E End(H), j2 -1, = turns H = M x H' into a
complex quaternionic vector bundle. Then F(H) M-4 H}, and
D,O 1(do
2
+ J * do)
is a holomorphic structure.
Example complex quaternionic

15. If L is line bundle and 0 E F(L) has no
a
zeros, then exactly one holomorphic structure

there exists D on (L, J) such
that 0 becomes holomorphic. In fact, any 0 E r(L) can be written as'O Op =
with p : M-* H, and our only chance is
1
Do :=
2
(Odp + JO * dy). (4.4)
This, indeed, satisfies the definition of a holomorphic structure.
Example 16. f : M-+ H is a conformal surface with left normal vector N,

If
then N is a complex structure for L Mx H, and there exists a unique D such
=
that D1 = 0. A section 0 lp is holomorphic if and only if dp + N * dp

=
0, =
i.e.
*dl-t =
Ndp.
The holomorphic sections are therefore the conformal maps with the same
left normal N as f .
In this case dim HO(L) ! 2, since I and f are independent
in HO(L).
Theorem 1. If L C H = M x H'+1 holomorphic is

curve a with complex
structure J, then the dual bundle L-1
a complex inherits
structure defined
by Jw := wJ. The pair (L-', J) has a canonical holomorphic structure D
characterized by the following fact: Any quaternionic linear form W : Hn+1 -4
H induces a section WL E V(L-') by restriction to the fibres of L. Then for
all w
DWL:::::: 0-
Proof. The vector bundle Lj- with fibre L-L

X
=
fw E (EP+I)* I WJL , =
0} has
a total space of real dimension 4n + 2. Therefore there exists W such that WL
has no zero. Example 15 yields a unique holomorphic structure D such that
DWL = 0. Now any a E F(L-') is of the form a WLIX for some /\: M-+ H.
=
Then, by (4.4), for any section 0 E r(L) we have
1
< Da,O >= < WLdA+JWL *dA,o >
2
(< wdA,o > + < w *dl\,Jo >)

2
I 1
(d < wA,O > + *d < WA, J'O >) -
< wA, do + *d(JO) >
2 2
1 I
(d < a,O > + * d < a, JO >) -
< wA, do + *d(JO) >
2 2
22 4 Vector Bundles
Note that *J = 6J implies do + *d(JV)) E F(L), and this allows us to replace

wA by a in the last term as well:
< Da, 0 >= -(d < a, > + * d < a, JO >) -

< a, do + *dJO >
2 2
This contains no reference to w, hence D is independent of the choice of W
such that WL has no zero. But the last equality shows Da = 0 for any a =
WL
with w E (Hn+l)*.
Remark 4. As we shallsee in the next section, a holomorphic curve L in
ffffP1 carries a natural holomorphic structure. In higher dimensional projec-
tive spaces this is no longer the case. Therefore L-1 rather than L plays a
prominent role in higher codimension.
5 The Mean Curvature Sphere
5.1 S-Theory
Let Mbe a'Riemann surface. Let
H:= Mx 19
denote the product bundle over M, and let S: M-+ End(EV) E F(End(H))
with S2 = -1 be a complex structure on H. Wesplit the differential according
to type:
do = do + d"O,
where cP and d' denote the Glinear and anti-linear components, respectively:
*d' =
Sd, *d" = -Sd".
Explicitly,
do (do -
S* do), d"O (do + S * dio).
2 2
So d" is holomorphic
a structure on (H, S), while d' is an anti-holomorphic
structure, a holomorphic
i.e. structure of (H, -S).
In general d(SO) i4 Sdo, and we decompose further:
d=,9+A, d"=a+Q,
where
a(SO) S90' 5(so) =

Sao,
AS -SA, QS =
-SQ.
For example, we explicitly have
I
(d"O -
Sd'(So)).
2
Then 0 defines a holomorphic structure and 0 an anti-holomorphic structure

on H, while A and Q are tensorial:

24 5 The Mean Curvature Sphere
AEr(KEnd-(H)), QEI(kEnd-(H)). (5.1)
For M-+ IV E F (H) we have, by definition of dS,
(dS)O =
d(SO) -
SdO
=
(,g + A)SO + (5 + Q)SO -
S(o9 + A)O -
S(5 + Q)0
=
ASO+ QSo -
SAO -
SQO
=
-2S(Q + A)O
=
2(*Q -
*A)O.
Hence
dS =
2(*Q -
*A), *dS =
2(A -
Q). (5.2)
Then
SdS =
2(Q + A),
whence conversely
Q (SdS -
*dS), A (SdS + *dS). (5-3)

4 4
Remark 5. Since A and Q are of different type, dS 0 if and only if A 0 = =
and Q 0. If dS
= =
0, then the i-eigenspaces of the complex endomorphism
S decompose H =
(M x C) ED (M x C). Therefore A and Q measure the
deviation from the "complex case".
5.2 The Mean Curvature Sphere
Wenow consider an immersed holomorphic curve L C H in HP1 with deriva-

tive J JL E S?'(Hom(L, HIL)). Then there exist complex structures J on
L and on HIL such that
*J = jj
We want to extend J and j to a complex structure of H, i.e. find an
S E F(End(H))
such that
SL =
L, SIL =
Ji 7rS = j7r-
Note that this implies

5.2 The Mean Curvature Sphere 25
7rdS(O) =
7r(d(SO) -
SdO) =
JJO -
ho =
0,
and therefore
dSL c L. (5.4)
The existence of S is'clear: Write H = L E) L' for some complementary
bundle L. Identify L' with HIL using 7r, and define SIL := Ji SIP := j.
Since L' is not unique, S is not unique. It is easy to see that S + R is
another such extension if and only if R: M-+ End(H) satisfies
RH c L c kerR,
whence R2 =
0, and
RS + SR = 0.
Note that R can be interpreted as an element of Hom(H/L, L). Then RV R.

We compute Q:
I((S
4
+ R)d(S + R) -
*d(S + R))
1(SdS
4
-
*dS) +
I
4
(SdR + RdS + RdR -
*dR)
1
Q+ (SdR + RdS + RdR -
*dR).
4
If V) E F(L), then
0 =
d(RO) =
dRO + Rdo,
RdRO = -R 2do = 0
and, by (5.4),
R dSO = 0
We can therefore continue
I 1
00=QO+ (SdRO -
*dRO) =
QO+ (-SRdo + *Rdo)
4 4
I
QO+ (-SRJO + R* JO) =
Q0 + (-SRJO + Rj JO).
4
=RS=-SR
Hence, for 0 E F(L),
00 =
Q0 -
-SRJO.
2
Now we start with any extension S of (J, J) and, in view of (5.5), define
R= -2SQ(X)6(X)-'7r: H -+ H (5.6)
for some X 54 0. First note that this definition is independent of the choice
of X 54 0. In fact, X F-+ R is positive-homegeneous of degree 0, and with
c =
cosO, s = sinO
Q(cX + SJX) (J(CX + SJX)) -') =

Q(X) (CI + SS) (J(X) (cl + ss))
-1
Q(X)6(X)-'.
Next
RS =
-2SQ(X)6X-'7rS =
-2SQ(X)6x-lj7r
=
-2SQ(X)SJj1-7r =
2S2Q(X)6 X1jr
= -SR.
By definition (5.6)
L c kerR,
and from (5.3) and (5.4) we get
L D I(SdS
4
-
*dS)L =
QL,
whence
RH c L.
Wehave now shown that S + R is another extension. Finally, using

(5.5), we find for 0 E F(L)
00 =
Q0 -
-1SRdo
2
=
Q0 -
I
2
S(-2SQ6-'7r)do
=
QV) -
QJ-lirdo = 0.
This shows
Theorem 2. Let L C H = M x EV be a holomorphic curve immersed into

HP'. Then there exists a unique complex structure S on H such that
SL =
L, dSL C L, (5.7)
*6 = 6 0 S = S 0 6, (5-8)
QJL = 0- (5.9)
5.3 Hopf Fields 27
S is a family 2-spheres, a sphere congruence in classical

of terms. Because
SpLp =
Lp the sphere Sp goes through Lp E HP', while dSL C L (or,
equivalently, JS S6) implies it is tangent to L in p, see examples
= 9 and 12.
In an affine coordinate system I,]

1
= L the sphere Sp has the same mean
curvature vector as f : M-4 R4 = H at p, see Remark 9. This motivates the
Definition 5. S is called the mean curvature sphere (congruence) of L. The

differential forms A, Q E S?'(End(H)) are called the Hopf fields of L.
Remark 6. Equations (5-7), (5.8) imply do + S * do E F(L) for 0 E F(L),
whence d" = 0 + Q =
12 (d + S * d) leaves L invariant. Hence an immersed
holomorphic curve in HP is a holomorphic subbundle of (H, S, d") and, in
particular, is a holomorphic quaternionic vector bundle itself.
Example 17. Let S E End(H2), S2 = -I. Then
S'= 11 E HP1 I Sl =
11 C HP1
is a 2-sphere in KPI. Let L denote the corresponding line bundle and endow
S' with complex the structure inherited from the immersion. Then the mean
curvaturesphere congruence of L is simply the constant map S' -+ Z of value

S: Wehave SL L by definition, = and the constancy implies dSL =
10} C L
and Q !(SdS
4 *dS)
= 0. -
=
5.3 Hppf Fields
In the following we shall frequently encounter differential forms. Note that

the usual definition of the wedge product of 1-forms
w A O(X, Y) =
w(X)O(Y) -
w(Y)O(X)
can be generalized verbatim to forms wi E 01 (Vi) with values in vector spaces
or bundles Vi, provided there is a product V, x V2 -+ V. Examples are the
composition End(V) x End(V) -+ End(V) or the pairing between the dual
V* and V.
On a Riemann surface M, any 2-form a E S22 is completely determined
by the quadratic form o-(X, JX) =: u(X), and we shall, for simplicity, often
use the latter. As an example,
w A O(X, JX) =
w(X)O(JX) -
w(JX)O(X)
will be written as
wAO=w*0-*wO. (5.10)
We now collect some information about the Hopf fields and the mean
curvature sphere congruence S : M-+ Z.

Lemma4.
d(A+Q) =
2(QA Q +AAA).
Proof. Recall from (5.2)

SdS =
2(A + Q).
Therefore, using AS =
-SA, QS =
-SQ,
d(A + Q) = Id(SdS)
2
=
1
2
(dS A dS)
=
2S(A Q) + Q) A S(A +
=2(AAA+AAQ+QAA+QAQ).
But A A Q = 0 by the following type argument: Using that
A is "right ff", and Q "left k))
we have
AAQ = A* Q -
*AQ =
A(-SQ) -
(-AS)Q = 0. (5-11)
Similarly QA A =
0, because A is left K and Q is right K.
Lemma5. Let L C H be an immersed surface and S a complex structure on
H stabilizing L such that dSL C L. Then QJL = 0 is equivalent to AH c L.
Notice that the kernels images of the 1-forms A and Q are well-defined:
and
if QxO = 0 for some also QjXO
X E TMthen -SQxO 0, and thus = =
QzO 0 = for any Z E TM. In other words, the kernels of Q and A are
independent of X E TM. The same remark holds for the respective images.
Proof. Wefirst need a formula for the derivative of 1-forms w E Q'(End(H))

which stabilize L, i.e., wL C L. If 7r =
*7rL, then for 0 E F(L)
7r (dw (X, Y) 0) =7r (d(wo) (X, Y) + w A dV) (X, Y))

=7r(x -
(W(Y)O) -
Y -
(W(X)O) -
W([X, YDO
.11
%vo
EF(L)
+ w (X) do (Y) -
w (Y) do (X))
=6(X)w(Y)O -
J(Y)w(X)O 7rw(X)dO(Y) + -
7rw(Y)dO(X)
=6(X)w(Y),O -
J(Y)w(X)O +,7rw(X)JO(Y) -
7rw(Y)JO(X)
=(J A w + -7rw A J)(X, Y)O,
where we wedge
composition. Note that the composition
over 7rWJ makes
sense, because w (L)
L, and L is annihilated by 7r. We apply this to A and
C
Q. Since AL C L, QL C L we have on L, by lemma 4,

5.4 The Conformal Gauss Map 29
0 = I7r(QA
2
Q +AAA) =
7r(dA+dQ)
= 6 A A + 7rA A 6 + 6 A Q + 7rQ A 6.
By a type argument similar to (5-11), we get J AA = 0 = -7rQ A 6. Further,
7rA A 6 = 7rA * 5 -
7r * AJ
=
-2S7rA6,
and similarly for the remaining term. Weobtain --xSA6 =

SJQIL or
-7rA6 =
JQIL -
-
Since AL C L and J(X) : L -+ HIL for X 54 0 is an isomorphism, we get

irA = 0 QJL = 0-
5.4 The Conformal Gauss Map
Definition 6. For a quaternionic vector space or bundle V of rank n and

A E End(V) we define
< A >: = traceR A,

Tn
where the trace is taken of the real endomorphism A. In particular < I >= 1.
We obtain an indefinite scalar product < A, B >:=< AB >.
Example 18. For A =

(a) with a =
ao + ial + ja2 + ka3 E H we have
< A >= 14ao =

ao,
4
and
< AA >= Rea2 = a20 _

a21 _a
2
2
_a23
Proposition 4. The mean curvature sphere S of an immersed Riemann sur-
face L satisfies
< dS, dS >=< *dS, *dS >, < dS, *dS >= 0,
i.e. S : M-+ Z is conformal.
Because of this proposition, S is also called the conformal Gauss map, see
Bryant [1].
Proof. Wehave QA =
0, and therefore
< Q,A >=< A, Q >= 0. (5.12)
Then, from (5.2),
< dS, dS >=4 < -S(Q + A), -S(Q + A) >= 4 < Q + A, Q + A >
=4 < Q -
A, Q -
A >=< *dS, *dS >
Similarly,
< dS, *dS >=4 < -S(Q + A), A -
Q>
=4(< SQQ> -
< S QA > + < SAQ > -
< SAA >).
=0
But, by a property of the real trace,
SAQ > =< QSA>=< -SQA >= 0,

SQQ> =< QSQ>=< -SQQ >= 0,
< SAA > =< ASA >=< -SAA >= 0.
6 Willmore Surfaces
Throughout this section Mdenotes a compact surface.
6.1 The Energy Functional
The set
Z =
IS E End(V) I S2 =
_jj
of oriented 2-spheres in RP1 is a submanifold of End(ffV) with
TsZ =
IX Fnd(EV) I XS -SX},
E
S -Z =
JY E End(fff) I YS SY}.
Here we use the (indefinite) inner product
1
< A, B >:=< AB >= -
traceR(AB)
8
defined in Section 5.3.
Definition 7. The energy functional of a map S : M --* Z Of a Riemann

surface M is defined by
E(S) :=
fm < dS A *dS >.
Critical points S of this functional with respect to variations of S are called

harmonic maps from M to Z.
Proposition 5. S is harmonic if and only if the Z-tangential component of

d * dS vanishes:
(d * dS)T = 0. (6-1)
This condition is equivalent to any of the following:

32 6 Willmore Surfaces
d(S * dS) =
o, (6.2)
d * A =
0, (6-3)
d * Q = 0. (6.4)
In fact,
d(S * dS) = 4d * Q = 4d * A =
S(d * dS)T =
(Sd * dS)T. (6-5)
Proof. Let St be a variation of S in Z with variational vector field Y.

Then SY -YS and
fm fm
d d
Wt- E(S)
< dS A *dS >= < dY A *dS > + < dS A *dY >.
Wt-
Using the wedge formula (5.10) and traceR(AB) =

traceR(BA), we get
< dS A *dY > =< dS(-dY) -

*dS * dY >=< dY A *dS > .
Thus
f' fm fm
d
Wt- E(S)
= 2 < dY A *dS >= -2 < Yd * dS >= -2 < Yd*dS >.
JM
Therefore S is harmonic if and only if d * dS is normal.

For the other equivalences, first note
0 = d * d(S') =
d(*dSS + S * dS)
=
(d * dS)S -
*dS A dS + dS A *dS + Sd * dS
-2(dS)2 -
2(*dS)2 + (d * dS)S + Sd dS
2dS A *dS + (d * dS)S + Sd * dS.
Now, together with *Q -

*A WSand A
2
=
1
4 (SdS + *dS), this implies
8d * Q 8d * A =
2d(S * dS)
2dS A *dS + 2Sd * dS
-(d dS)S + Sd * dS
S(d dS + S(d * dS)S).

=2(d*dS)T
We now consider the case where S is the mean curvature sphere of an
immersed holomorphic curve. We decompose dS into the Hopf fields.
Lemma6.
dS A *dS >= 4(< A A *A > + < Q A *Q >), (6-6)

dS A SdS >= 4(< A A *A > -
< Q A *Q >). (6.7)
6.1 The Energy Functional 33
Proof. Recall from section 5.1
dS ='2(*Q -
*A), *dS =
2(A -
Q), SdS =
2(Q + A).
Further
*QAA=O, *AAQ=O
by type. Therefore
< dS A *dS > = 4 < (*Q -
*A) A (A -
Q) >
=-4<*QAQ> -4< *AAA>

= 4 < Q A *Q > +4 < A A *A >,
and similarly for < dS A SdS >.
Lemma7. Let V be a quaternionic vector space, L CV a quaternionic line,

S, B E End(V) such that
S2 _j T, SB =
-BS, image B C L.
Then
2
traceR B < 0,
with equality if and only if BIL = 0-
Proof. We may assume B 54 0. Then L BV, and SB -BS implies

SL = L. Let 0 E L\101, and
So =
OA, BO OIL.
Then 'X2 =-1, and BS = -SB implies
AP -PA.
Therefore p is imaginary, too. It follows B20 =

_JpJ2 0, and
traceR B2 =
traceR B2 IL =
-41p 12.
This can be applied to A or Q instead of B, since their rank is < 1. We
obtain
Lemma8. For an immersed holomorphic curve L we have
I
< A A *A >= < AIL A *AIL >i (6.8)
2
and
<AA*A>>O, <Q.A*Q>>O. (6.9)

In particular E(S) > 0.
Proof,
< AA *A >
8
traceR(-A' -
(*A)' traceR A
2
4
=-ASSA=A2
Because dim L
2
dim H we similarly have
12
< AIL A *AIL >= -
traceR A
IL7
2
see section 5.3. Because AH c L, we have
traceR A 12L
2
traceR A =
This proves (6.8). The positivity follows from Lemma7.
Proposition 6. 1. The (alternating!) 2-form w E S?2 (Z) defined by
ws(X,Y)=<X,SY>, forSEZ,X,YETsZ,
is closed.
2. If S : M-* Z, and dS =
2(*Q -
*A) as usual, see section 5.1 (5.3), then
S*w = 2 < A A *A > -2 < Q A *Q >.
In particular,
f
1
degS:= <AA*A>-<QA*Q>
7r M
A
is a topological invariant of S.
Remark 7. S maps the surface Minto the 8-dimensional

Since Z, deg S cer-
tainly mapping degree of S. But for immersed holomorphic curves
is not the
it is the difference of two mapping degrees deg S deg N deg R, where = -
N, R : M-4 S2 are the left and right normal vector in affine coordinates, see
chapter 7.
Proof. (i). Weconsider the 2-form on End(EP) defined by

1
COS(X, Y) := -(< X, SY > -
< Y, Sx >).
2
Then dsCo (X, Y, Z) is a linear combination of terms of the form
< Y'XZ >
But if X, Y, Z E TsZ, S E Z, we get

6.2 The Willmore Fanctional 35
< y )XZ > < S2yXZ >=< SyXZS >
=< S2yXZ >= _

< y IXZ >'
hence < Y, XZ > = 0. Therefore, if t : Z -+ End(ffR2) is the inclusion,
dw = dt*Cv = t*dCo = 0.
(ii). Wehave
S*w(X, Y) =< dS(X), SdS(Y) >
1
(< dS(X)SdS(Y) > -
< SdS(X)dS(Y) >)
2
1
(< dS(X)SdS(Y) > -
< dS(Y)SdS(X) >)

2
I
= < dS A SdS > (X, Y),
2
and Lemma6 yields the formula.

The topological invarlance under deformations of S follows from Stokes
theorem: If 9 : Mx [0, 1] 3 deforms So: M-4 3 into S1, then
0
fmx[0 , 1]
dg*w
f MAx 1
g*"'
J MX0
g*W
fm S*1W-
fm so* W.
Remark 8. From
E(S) = 4
fm < A A *A > + < Q A *Q >
= 8
fm < A A *A > +
4fm (< Q A *Q > -
< A A *A >)
topological invariant
we see that for variational problems the energy functional can be replaced by
the integral of < A A *A >.
6.2 The Willmore Functional
Definition 8. Let L be a compact immersed holomorphic curve in HP' with

Hopf field A. The Willmore functional of L is defined as
W(L) :=
-I
7r
f M
A
< A A *A >
If we vary the immersion L : M --+ HP, it will in general not remain a
holomorphic curve. On the other hand, any immersion induces a complex

structure J on Msuch that with respect to this it is a holomorphic curve, see
Proposition 3. Critical points of Wwith respect to such variations are called

Willmore surfaces. If we consider only variations of L fixing the conformal
structure of M they are called constrained Willmore surfaces, but we shall
not treat this case here.
Example 19. For immersed surfaces in R4 we have
W(L)
47r f Am
A
(H
2
-
K -
Kj-) ldf I 2
see section 7.3, Proposition 13.
Theorem 3 (Ejiri [2], Rigoli [12]). An immersed holomorphic curve L is

Willmore if and only if its mean curvature sphere S is harmonic.
Proof. Let Lt be a variation, and St its mean curvature sphere. Note that for
Lt to stay conformal the complex structure, i.e. the operator *, varies, too.
The variation has variational vector field Y E F(Hom(L, HIL))
a
given by
d
Y,O:= 7r( 0), Ot E F(Lt).
dt
t=0
As usual, we abbreviate t=o by a dot Note that for E F (L)

dt .
7r, o =
7r(SO)* -
irS =
YSO -
57r =
(YS -
SY)O. (6.10)
Wenow compute the variation of the energy, which is as good as the Willmore
functional as long as we vary L. By contrast, in the proof of Proposition 5
the conformal structure on Mwas fixed, and no L was involved.
t=o f
d d
E(St) < dSt A *tdSt >
dt dt M
A
t=0
fm < d, A *dS > +

fm < dS A dS > +
fm < dS A *d >.
In general < A A *B >=< B A *A >, because traceR(AB) =

traceR(BA).
Hence
(6-11)
Next we claim
II = 0. (6-12)
6.2 The Willmore Functional 37
On TMlet B, i.e. w(X) =: w(BX). Then we have BJ + JB =

0, and
< dS A dS > (X, JX) =< dS(X) dS(JX) > -
dS(JX)MS(X) >
<
=< dS(X)dS(BJX) > -

< dS(JX)dS(BX) >
= -
< dS(X)dS(JBX) > < dS(BX)dS(JX)
-
>.
But S is conformal, see Proposition 4, therefore
< dS (X) dS (JX) > = 0 for all X.
Differentiation with respect to X yields
< dS(X)dS(JY) > + < dS(Y)dS(JX) >= 0
for all X, Y. Using this with Y = BX we get (6.12).

Now, we compute the integral I.
I
fm < ,d*dS >
(6.5)
4f M
A
< ,Sd*Q>
2 f M
A
traceR (, Sd * Q).
We shall show in the following lemma that
imaged * QCL c ker d * Q.
Therefore we can consider d * Q as a 2-form
d * Q Ep2 (Hom(H/L, L),
and continue
fm (, Sd Q H)
1
1 traceR * : H -+
2
fm (7r Sd Q HIL HIL)

1
traceR * : -+
2
i
2 fm traceR(7r JLSd Q HIL HIL)
,
* : --+
(6.1o)
1f traceR((YS-SY)(Sd*Q):H1L-4H1L)
2 M
A
fm traceR(Yd Q) I traceR(SYSd
I
* -
* Q).
2 2 M
Now,d Q is tangential by (6.5), and hence anti-commutes with S. Thus

1=: -
2 fm traceR(Yd Q) * +
2 f M
A
traceR(SYd QS)
fm traceR(Yd Q) *
-8
fm Yd*Q < >
Wetherefore showed
E(St)=-8f
d
<Yd*Q>.
Tt t=O M
A
Since Y (-= Q2 (Hom(H/L, L), this vanishes for all variational vector fields Y
if and only if
d*Q = 0.
In the proof we made use of the following
Lemma9.
imaged * QCL C kerd * Q.
Proof. Foro E F(L)
0 =
d(*Qo) -
(d * Q)o -
*Q A do =
(d * Q)o -
*Q A do,
because QJL = 0. But *Q is right K, and 6 is left K. Hence, by type,
(d * Q)o =
*Q A 60 = 0.
This shows the right hand inclusion. Also,
7r (d * Q) (X, JX) = 7r (d * A) (X, JX)

=
7r(X -
(*A(JX)) -
JX -
(*A(X)) -
*A([X, JX]))
L-valued
=
J(X) * A(JX) -
J.(JX) * A(X)
=
-J(X)A(X) -
6(X)SSA(X)
=0. -
7 Metric and Affine Conformal Geometry
Weconsider the metric extrinsic geometry of f : M-+ R in. relation to the
quantities associated to
L:= M HPI.
For brevity we write < .,.

> instead of < >R-
7.1 Surfaces in Euclidean Space
Let N, R denote the left and right normal vector of f : M H, i.e.
*df =
Ndf =
-df R.
Proposition 7. The second fundamental form II(X, Y) (X df (Y))--L of

f is given by
II(X, Y) (*df (Y)dR(X) -
dN(X) * df (Y)). (7.1)

2
Proof. We know from Lemma 2 that v -+ N(x)vR(x) is an involution with

the tangent space as its fixed point set:
Ndf (Y)R =
df (Y) (7.2)
Its (-I)-eigenspace is the normal space, so we need to compute
1
II (X, Y) =
(X df (Y)
- -
NX -
df (Y) R).
2
But differentiation of (7.2) yields
dN(X)df (Y)R + NX -
df (Y)R + Ndf (Y)dR(X) = X -
df (Y),
or
X -
df (Y) -
NX -
df (Y)R =
dN(X)df (Y)R + Ndf (Y)dR(X)
=
-dN(X) * df (Y) + *df (Y)dR(X).

40 7 Metric and Affine Conformal Geometry
Proposition 8. The mean curvature vector 'H

2
trace II is given by
'Rdf (*dR + RdR), dfR (*dN + NdN). (7-3)

2 2
Proof. By definition of the trace,
4'H jdfJ2 =
*dfdR -
dN * df -
df * dR + *dNdf (7.4)
=
-df (*dR + RdR) + (*dN + NdN)df, (7.5)
but
(*dN + NdN)df =
*dNdf -
dN * df = -dN A df =
-d(Ndf)
=
-df A dR =
-df (*dR + RdR).
If follows that
27ildfI2 =
-df (*dR + RdR),
and
2 ldfTf dR + dRR)Tf =
(*dR + RdR)Tf
Similarly for N.
Proposition 9. Let K denote the Gaussian curvature of (M, f *

< >R)
and let K' denote the normal curvature of f defined by
Kj- :=< Rj- (X, JX) , N >R)
where X E TpM, and E-Lp M are unit vectors. Then
1
KIdf 12 =
(< *dR, RdR > + < *dN, NdN >) (7.6)
2
1
ldf 12
-1 (< *dR, NdN >)
K =
2
RdR > -
< *dN, (7.7)
Proof.
Kldfl4(X) =< II(X, X), I.T(jX, jX) > _III(X, jX)12.

Therefore
7.1 Surfaces in Euclidean Space 41
4KIdf 14 =< *df dR -

dN * df, -df * dR + *dNdf >
-
< *df * dR -
*dN * df, -df dR + dNdf >
= < N (df dR + dNdf ), -df * dR + *dNdf >
< N(df * dR + *dNdf), -df dR + dNdf >
< df dR + dNdf, N(-df * dR + *dNdf) >
< df * dR + *dNdf, N(-dfdR + dNdf) >
=- < dfdR + dNdf, dfR * dR + N * dNdf >
+ < df dR + *dNdf, dfRdR + NdNdf >
< dfdR, dfR * dR > -

< dfdR, N dNdf >
< dNdf, df R * dR > -

< dNdf, N dNdf >
+ < df * dR,dfRdR > + < df * dR,NdNdf >
+ < *dNdf, dfRdR > + < *dNdf, NdNdf >
ldf 12 < dR, RdR > -
< dfdR, N* dNdf >
+ < dNdf, Ndf dR > -ldf 12 < dN, N * dN >
+ ldf 12 < *dR, RdR > + < df * dR, NdNdf >
< *dNdf , Ndf dR > +jdf 12 < *dN, NdN >
I dyl2(< dR,R*dR > + < dN,N* dN >
< *dR, RdR > -

< *dN, NdN >)
21dfI2 (< dR,R * dR > + < dN,N * dN >).
This proves the formula for K. Using (7.1) and the Ricci equation
Kj- =< N II(X, JX), II(X, X) -
II(JX, JX) >,
we find, after a similar computation,
jdfJ2
1
4K =< *dR -
RdR, RdR > -

< *dN -
NdN, NdN >

+ < df (*dR -
RdR), NdNdf > -

< (*dN -
NdN)df, dfRdR >
On this we use (7.5) to obtain (7.7).

As a corollary we have
Proposition 10. The pull-back of the 2-sphere area under R is given by
R*dA =< *dR, RdR >
Integrating this for compact M yields
f
1
T7r Kjdf 12 (deg R+ deg N).
M
A 2
In 3-space (R =
N) this is a version ofthe Gauss-Bonnet theorem.
Proposition 11. We obtain
(J-H12 -
K -
Kj-) ldf12 4
1 * dR -
RdRJ2
In particular, if f : M-+ Im H = R' then Kj- =

0, and the classical Willmore
integrand is given by
(I Ij 12 -
K)Idfl2 1 * dR -
RdRJ2. (7.8)
4
Proof. Equations (7.3), (7.6), (7.7) give
(I,HI2 -
K -
K-L)Idfl2 11
4
* dR + RdRj2_ < *dR, RdR >
11
4
* dR12 + 1IRdRI2
4 2
< *dR, RdR >
1 2
41 * dR -
RdRI
7.2 The Mean Curvature Sphere in Affine Coordinates
We now discuss the characteristic properties of S in affine coordinates. We
describe S relative to the frame

0 1
i.e. we write S = GMG-1, where
G
01 f)
First, SL C L is equivalent to S EV -- H2 having the following matrix
representation:
S=
(1f) (' -R)
01 -H
0
0 1f) (7.9)
where N, R, H : M-+ H. From S2 = _I
N2=-l=R
2
,
RH HN. (7.10)
The choice of symbols is deliberate: N and R turn out to be the left and right
normal vectors, of f while H is closely
,
related to its mean curvature vector
X
The bundle L, has the nowhere vanishing section (fl) E V (L). Using this
section, we compute
7.2 The Mean Curvature Sphere in Affine Coordinates 43
(f) =
,
(*df) 0
ird(S (f 7rd( (f 7r((-df R) (f ) R)

-df
is =
R)) =
0
+
1
(-dR)) -7r
0
sj
(f)
1
= irSd
(f) Ir((Ndf)1 0
+
(f) (-Hdf))
1 (Ndf = Ir
0
Therefore *6 = S6 = JS is equivalent to
*df =
Ndf =
-df R,
and we have identified N and R.

For the
computation of the Hopf fields, we need dS. This is a straight-
forward but lengthy computation, somewhat simplified by the fact that
GdG dG =G-dG. We skip the details
= and give the result:
( -dfH + dN -dfR Ndf)

-
dS = G
-dH -dR + Hdf P7 '
SdS = G
(HdfH -NdfH
+ RdH
+ NdN
-
HdN Hdf R + R
0
dR) G-1.
Romthis we obtain
Q = SdS -
*dS
( NdN *dN 0
*dR) G-1
-
= G
*dH + HdfH + RdH -
HdN 2HdfR + RdR +
4A = SdS + *dS
( 2Ndf
*dR)
NdN + *dN H 0
G-1.
-
G
.
-
* dH + HdfH + RdH -
HdN RdR -
The condition QIL =

0, and the corresponding AH C L, which we have not
used so far-, have the following equivalents:
2Hdf dR -
R* dR, (7.11)
2dfH dN -
N * dN. (7.12)
Together with equations (7.3) we find
2Hdf = dR -
R dR =
-R(*dR + RdR) -2RTtdf,
=
2dfH = dN -
N dN =
-N(*dN + NdN) 2NdfR = =
-2dfRR,
and therefore
H= -RN = -RR. (7.13)

Remark 9. Given an immersed holomorphic curve L the mean cur-
vature vector of f at x E M is determined by Sx. On the other hand, S"

is the mean curvature sphere of Sx, see Example 17. Therefore S,' and f
have, in fact, the same mean curvature vector at x, justifying the name mean
curvature sphere.
Equations (7.11), (7.12) simplify the coordinate expressions for the Hopf
fields, which we now write as follows
Proposition 12.
( 0) G-1,
dN + N * dN 0
4*Q=G -2dH + w
(7.14)
4*A= G
(w 0
dR + R *
0
dR) G-', (7.15)
where G
(01), f
and w = dH + H * dfH + R * dH -
H * dN.
Using (7.12) we can rewrite
w = dH + R * dH + 1H(NdN
2
-
*dN).
Proof. Weonly have to consider the reformulation of w. But
H* dfH -
H * dN IH * (dN -
N* dN) -
H * dN
2
-
1H* (dN+N*dN) H(NdN -
*dN).
2 2
7.3 The Willmore Condition in Affine Coordinates
We use the notations of the previous Proposition 12, and in addition abbre-
viate
v, = dR+R*dR.
Note that
V = -dR +. *dRR -- -dR -

R * dR = -v.
Proposition 13. The Willmore integrand is given by

1 1
< A A *A > =
16
JRdR -
*dR12 =
4
(IHI2 -
K -
K-L)JdfJ2.
For f : M-4 R, this is the classical integrand
1
< A A *A >= (Ih 12 -
K)Idfl2.
4
7.3 The Willmore Condition in Affine Coordinates 45
Proof.
< A A *A > traceR(-A' -
(*A)) =
4
traceR(A2)
8
1
4
4 Re( 1V)
4
2
16
IV12 =
16
JdR + R* dR12
16
jRdR -
*dR12.
Now see Proposition 11 and, for the second equality, (7.8).
We now express the Euler-Lagrange equation d * A = 0 for Willmore
surfaces in affine coordinates. If we write 4 * A =
GMG-1, then
4d * A =
G(G-1 dG A M+ dM + MA G-1 dG)G-1,
and again using G-'dG = dG we easily find
4d * A = G
( df A
dw
w
dv +
df A
w
v
A df) G-1.
Most entries of this matrix vanish:
Proposition 14. We have
df Aw=O (7.16)
df Av=O (7.17)
dv+wAdf =-(2dH-W)Adf =0. (7.18)
Proof. We have
df A w =
df A dH + df A R * dH + Idf
2
A H(NdN -
*dN)
=df AdH+dfRA*dH+ IdfH

2
A (NdN -
*dN)
1
df A dH -
*df A *dH+ dfH A (NdN -
*dN),
2
io
but
*dfH =
df (-R)H -df HN
2
*(NdN -
*dN) =
(N * dN -
N dN) =
-N(NdN -
*dN).
Hence, by type, the second term vanishes as well, and we get (7.16).
A similar, but simpler, computation shows (7.17)
Next, using (7.11), we consider
dv+wAdf =d(dR+R*dR)+wAdf
=
d(-2Hdf) + w A df
=
(-2dH + w) A df
=
(-dH + R * dH + 1H(NdN -
*dN)) A df.
2 "o
Again we show *a =
aN, ON. Then .(7.18) will follow by type.
Clearly
*(NdN -
*dN) N * dN + NdNN= (NdN -
*dN)N,
showing *0 flN. Further
*a -
aN * dH -
RdH + dHN -
R(*dH)N
* dH -
d(RH) + (dR) H+ d(HN) -

HdN -
R(d(HN) -
HdN)
=RH =RH
+R2* dH + (dR)H -
HdN -
R* ((dR)H + RdH -
HdN)
(dR)H -
HdN -
R* (dR)H + RH* dN)
(dR -
R* dR)H -
H(dN -
N* dN)
2HdfH -
H(2dfH)
= 0.
As a corollary we get:
Proposition 15.
(dw 0) (dw -fdwf)

1 0 0 fdw
d*A= G G-1
4 4 -dw
with w = dH + R * dH + !H(NdN
2
-
*dN).
Therefore f is Willmore if and only if dw 0.
Example 20 (Willmore Cylinder). Let -y : RIm H be a unit-speed curve,

and f : R2 -+ H the cylinder defined by
f (S' t) =
-Y(S) + t
with the conformal structure J-L

as
= -2-.
at
Then using Proposition 15, we obtain,
after some computation, that f is (non-compact) Willmore, if and only if
Ir.3+ K11 -r.7, 2

=
0, (r,2-ol = 0.
2
This is exactly the condition that -y be a free elastic curve.

8 Twistor Projections
8.1 Twistor Projections
Let E C H := M x H' = Mx C' be a complex (not a quaternionic) line

subbundle over a Riemann surface M with complex structure JE induced
from right multiplication by i on IV.
We define JE E f2l (Hom(E, HIE)) by
JEO:= 7rEdo, 0 E F(E),
where 7rE : H -+ HIE is the projection.
Definition 9. E is called a holomorphic curve in CP1, if
*6E :-.,: 6E JE
This is equivalent to the fact that the holomorphic structure
d"O =
2
1(do + i * do) (8.1)
of H maps F(E) into itself, and hence induces a holomorphic structure on
the complex line bundle E.

A complex line bundle E C H induces a quaternionic line bundle
L=EH=EE)Ej CH.
The complex structure JE admits a unique extension to the structure of a

complex quaternionic bundle (L, J), namely right-multiplication by (-i) on
Ej. Conversely, a complex quaternionic line bundle (L, J) C H induces a
complex line bundle
E:= JOE L1JO=Oi}.
Definition 10. We call (L, J) the twistor projection of E, and E the twistor
lift of (L, J).

48 8 Twistor Projections
Remark 10. 'As in

quaternionic case, any map f : M -+ CP' induces
the
a complex line E, where the fibre over p, is f (p), and vice versa.
bundle
Holomorphic curves as defined above correspond to holomorphic curves in
the sense of complex analysis. The correspondence between E and (L, J) is
mediated by the Penrose twistor projection Cp3 -+ ffffpl.
Theorem 4. Let E C H be a a complex line subbundle over a Riemann

surface M, and (L, J) its twistor projection.
1. Then (L,J) is a holomorphic curve, i.e.
*6L ---:
JLJ) (8.2)
if and only if
2
(JE + *JEJE) E S?'(Hom(E, LIE)) C S?'(Hom(E, HIE)).
In this case we have a differential operator
F(L) -+ S?'(L),o DO:= T(do

2
+ *d(JO))
Its (1, 0) -part is given by
1
AL := (D + JDJ) E F(K End- (L)). (8.3)
2
2. If (L, J) is a holomorphic curve then
1(Je
2
+ *JEJe) =
7r-eALIE-
Moreover,
1(6E
2
+ *JEJE) = 0 4== AL = 0-
In other words: The twistor projections of holomorphic curves in Cp3 are
exactly the holomorphic curves in HP' with AL 0- =
3. Let L be an immersed holomorphic curve with mean curvature sphere

congruence S E F(End-(H)), and J =
SIL. Then
1
A
4
(SdS + *dS) E 1'(k End- (H))
satisfies
AIL = AL-
8.1 Twistor Projections 49
Proof, (i). If (L, J) is a holomorphic curve then, for any F(L),

I
-7rL(dO + *d(Jo)) = 0.
2
But then
2
1(do + *d(J'O)) E S?1 (L)
a fortiori for all c F (E). It follows
7rE (do + *d(jEo)) E 01 (LIE).

2
Conversely, 17rE (do + *d(JEO)) E 01 (LIE) for 0 E F(E) implies

2
1(do
2
+ *d(JEO)) E S?1 (L),
and therefore
*6LIE =
6LJIE-
Again for 0 E F(E)
I(d(oJ)
2
+ *d(Joj))
2
((do)j + *d(JO)j))
2
(do
N
+ *d(JEO))j E S?1 (L).
V
ES-23-(L)
This shows
*6L = 6LJ-
By the preceding, maps into 01 (L). Its (1, 0)-part is
1(-b
2
-
J
but for 0 E F(L)
*AO (*do -
d(Jo)) = -No.
2
This proves (8-3).

(ii). For 0 E F(L) we have
ALO =
4
I(do + *d(JO) + J(dJO -
*dV))) (8.4)
But for 0 E F(E) we have J(dJo -
*do) =
J(do + *doi)i, and hence
ALO 4((do + *d(JO)) + J(dO + *d(JO))i).
By assumption !(do
2
+ *d(Jo)) has values in L = E G Ej, and AL 0 is its
Ej-component, namely the component in the (-i)-eigenspace Of JIL. In par-
ticular,
irEALO =
IrE 1(do
2
+ *d(Jo))
2
(6E + *JEJ)O)
and 7rE(ALO) = 0 if and only if ALO = 0. Since ALIE determines AL by

linearity, 17rE(do
2 *d(oi))
+ = 0 =* AL = 0-
(iii). For E F(L)
A0 (SdS + *dS),o
4
1(S(d(So)
4
-
Sdo) + *d(SO) -
*Sdo)
I(S(d(So)
4
-
*dO) + *d(SO) + do).
Comparison with (8.4) shows AIL = AL.
In view of lemma 8 this implies the following
Corollary 1. The twistor projections of holomorphic curves in Cp3 are ex-
actly the holomorphic curves in HP1 with vanishing Willmore functional.
8.2 Super-Conformal Immersions
Given a surface conformally immersed into R4, the image of a tangential

circle under the quadratic second fundamental form is (a double cover of)
an ellipse in the normal space, centered at the mean curvature vector, the
so-called curvature ellipse. The surface is called super-conformal if this ellipse
is a circle.
If N and R are the left and right normal vector of f, then according to
Proposition 7 we have
II(X, Y) (*df (Y)dR(X) -
dN(X) * df (Y)),
2
and therefore
8.2 Super-Conformal Immersions 51
H(cos OX + sin OJX, cos OX + sin OJX)
(*df(cosOX+sinOJX)dR(cosOX+sinOJX)
2
-
dN(cos OX + sin OJX) * df (cos OX + sin OJX))

I(df
2
(cos OJX -
sin OX)dR(cos OX + sin OJX)
-
dN(cos OX + sin 0 JX) df (cos 0 JX -
sin OX))
1
(COS2 0(df (JX)dR(X) -
dN(X)df (JX))
2
-
sin20(df (X)dR(JX) -
dN(JX)df (X))
+ cos 0 sin 0(df (JX)dR(JX) -
df (X)dR(X)
+ dN(X)df (X) -
dN(JX)df (JX)).
Using COS2 0 =
.1
2 (1 + cos 20), sin
2
0 =!(I2 -
cos 20) we get
H(cos OX + sin OJX, cos OX + sin OJX)
=-(*df (X)dR(X) -
dN(X) * df (X) + *df (JX)d ?(JX) -
dN('JX) * df (JX))
4
=211(X,X) =211(3X,JX)
I
+ cos 20(df (JX)dR.(X) -
dN(X)df (JX) + df (X)dR(JX) -
d1V(JX)df (X))
4
+ sin 20(df (JX)dR(JX) -
df (X)dR(X) + dN(X)df (,Y). -
dN(JX)df (JX))
4
=7 Ild y (X) 12,
+ cos 20(df (X)(*dR(X) -
RdR(X)) -
(*dN(X) -
NdN(X))df (X))
4
=:a =:b
I
+ sin 20N(a + b).
4
This is a circle if and only if a -

b and N(a + b) are orthogonal and have
same length. This is clearly the case if a = 0 or b =
0, but these are in fact
the only possibilities: Assume that there exists P E H, p2 -1 with
=
N(a + b) =
P(a -
b), (8.5)
and note that
Na =
aR, Nb = bR.
We multiply (8-5) by N from the left or by R from the right to obtain
-(a + b) =
NP(a -
b), -(a + b) =
PN(a -
b)
respectively. Therefore (PN -
NP) (a -
b) =
0, which implies P =
J-N, and
hence a = 0 or b =
0, or a- b = 0. But then also a+b 0, whence
= a = b = 0.
It follows that the immersion is super-conformal if and only if
*dR(X) -
RdR(X) =
0, or * dN(X) -
NdN(X) = 0.
By preceding argument, this holds for a particular

the choice of X, but
then obviously follows for all X.
it
Wemention that f -+ I exchanges N and R, hence f is super-conformal,
if and only if *dR RdR 0 for f or for 1. In view of proposition
-
=
12, this
is equivalent to AIL 0, and by Theorem 4 we obtain:
=
Theorem 5. A conformally immersed Riemann surface f : M-+ H = RI is
super-conformal if and only if M

1
: M -+ ELO, or
If I
I
: M -+ HP' is the
twistor projection of a holomorphic curve in Cp3.

9 Micklund Transforms of Willmore Surfaces
In this section we shall describe a method to'construct new Willmore surfaces

from given one. The construction
a depends on the choice of a point 00, and
therefore generously offers a 4-parameter family of such transformations. On
the other hand, the necessary computations are not invariant, and therefore
ought to be done in affine coordinates.

The transformation theory is essentially local: This fact will be hidden
in the assumption that the transforms are again immersions. We shall also
ignore period problems.
9.1 Micklund Transforms
Let f : M-+ ffff be a Willmore surface with'N, R, H, and
w = dH+H*dfH+R*dH-H*dN.
Then
dw =
0',
and hence we can integrate it. Assume that g : M-+ H is an immersion with
dg = 1W. (9.1)
2
(Note that the integral of w/2 may have periods, so in general g is defined only on
a covering of M. Weignore this problem.)

We want to show that g is again a Willmore surface called a Bdcklund
transform of f . Using this name, we refer to the fact that in a given category
of surfaces we construct new examples from old onesby solving an ODE(9.1),
similar to the classical Biicklund transforms of K-surfaces, see Tenenblat [13].
We denote the symbols associated to g by a subscript (-)g, and want
to prove dw, = 0. The computation of wg can be done under the weaker
assumption (9.2), which holds in the case above, see Proposition 14.
Proposition 16. Let f , g : M-+ H be immersions such that

54 9 Micklund Transforms of Willmore Surfaces
df A dg =
Q. (9.2)
Then f and g induce the same conformal structure on M, and
Ng =
-R, (9.3)
dg(2dHg -
wg) =
-wdf. (9.4)
Proof. Define * using the conformal structure induced by f. Then
0 =
df A dg =
df * dg -
df (-R)dg,
which implies *dg =
-Rdg. Hence g is conformal, too, and Ng = -R.
For the next computations recall the equations (7.10), and (7.11), (7.12):
HN= RH, -
2dfH=dN-N*dN, 2Hdf =dR-R*dR,

w=dH+H*dfH+R*dH-H*dN.
Then
Rw =RdH + RH dfH -
*dH -
RH dN
=RdH + HN dfH -
*dH -
HN dN
=RdH -
HdfH -
*dH -
H(N * dN -
dN) -
HdN
=RdH -
HdN + HdfH -
*dH. (9.5)
With dRH + RdH = dHN + HdN this becomes
Rw dHN -
*dH -
dRH + HdfH. (9.6)
Next
2dgHg dNg -
Ng * dNg = -dR -
R * dR.
Therefore
-dg A dHg d(-dR -

R* dR) d(dR -
R* dR) = dH A df,
2 2
or
dg(*dHg + RgdHg) =
-(dHN -
*dH)df. (9.7)
We now use (9.5) and (9.7) to compute
Ngdg(2dHg wg) -
dgRg (2dHg
-
wg) -
=dg(-2RgdHg + RgdHg HgdNg + HgdgHg - -
*dHg)
-
dg (Rg dHg + *dHg) + dgHg (dgHg

-
dNg) -
=(dHN *dH)df + dgHg(dgHg

-
dNg) -
I
=(dHN -
*dH)df + (dNg -
Ng * dNg) ((dNg -
Ng * dNg) -
2dNg)
4
1
=(dHN -
*dH)df -
(dR + R* dR)(dR -
R* dR).
4
9.1 Bkklund Transforms 55
Similarly, using (9.6),
-Ngwdf =
Rwdf
=
(dHN -
*dH)df -
(dR -
Hdf)Hdf
=
(dHN -
*dH)df -1(2dR
4
-
dR + R * dR)(dR -
R* dR)
=
(dHN -
*dH)df -1(dR
4
+ R* dR)(dR -
R* dR).
Comparison yields (9.4).

If f is Willmore, and g is defined by (9.1), then
dg(2df + 2dHg -
wg) =
2dgdf + dg(2dHg -
wg) =
(2dg -
w)df = 0.
Hence
wg =
2d(f + Hg), (9-8)
and g is Willmore, too.
Now assume that h g -
H is again an immersion. Then, by Proposi-
tion 14,
2dh A df =
(2dg -
2dH) A df =
(w -
2dH) A df = 0.
Proposition 16 applied to (h, f ) instead of (f g)

,
then says
-whdh =
df (2dH -
w) =
df (2dH -
2dg) =
-2df A.
We find Wh =
2df, whence h is again a Willmore surface. We call g a for-
ward, and h a backward Bdcklund transform of f. h can be obtained without
reference to 9 by integrating d(g -
H) = 1w -
dH.
2
Note that f is a forward Bdcklund transform of h because df = !Wh,
2
and
is also a backward transform of g because df =
!wg
2
-
dHg, see (9.8).

The concept of Bdeklund transformations depends on the choice of affine
coordinates. The following theorem clarifies this situation. ,
Theorem,6. Let L be a Willmore surface in RP'. Choose non-zero 0 E
(EV)*, a E IV such that < 0, a >= 0. Then
d < P, *Aa >= 0 = d < 0, *Qa > .
If g, h : M-+ H C HP' are immersions that satisfy
dg=2<0,*Aa>, dh=2<0,*Qa>,
they are again Willmore surfaces, called forward respectively backward Micklund
transforms of L. The free choice of 0 implies that there is a whole S' of such
pairs of Bdcklund transforms. (Different choices of a result in Moebius trans-
forms g -+ gA, or h -+ hA, for a constant A.)

56 9 Bhcklund 'hansforms of Willmore Surfaces
Proof. Choose b E H2, a E (IV)* such that a, b and a, fl are dual bases. Then
2 < P, *Aa >=

I
W'. 2 < #,*Qa >= 1w -
dH,
2 2
see Proposition 12.
We can now proceed from g with another forward Bdcklund transform.

To do so, we must integrate 'Fw
2 -9 d(f + Hg). But, up to a translational
constant, this yields
f f + Hg. (9.9)
We now observe
Lemma10.
fil) E ker A.
Proof. Note that ker A = ker *A. By Proposition 12 we have
4*A
( )(
I
0 1
f
w
0
) ( _f) (f Hg)
dR + R * dR
0
0 1
+
1
( if) (0
01 w dR+R*dR) Hg)
0
1
(0 1)
1 f
wHg+ dR+R*dR
=-dN,+Ng*dN.
V
( ) ( 2dgHg
1
0 1
f 0
-
2dgHg ) = 0.
Similarly the twofold backward Bdcklund transform f satisfies
f )H D image Q.
But this means that away from the zeros of A orQ the 2-step Backlund
transforms of a Willmore surface L in HP1 can be described simply as L
ker A or L =
image Q. In particular there are no periods arising.
Weobtain a chain of Bdcklund transforms
-+j-+ h -4f -+ g -+f
L -+ L -+
Of course, the chain may break down if we arrive at non-immersed surfaces,

or it may close up.
9.2 Two-Step Bdcklund 'kansforms 57
9.2 Two-Step Micklund Transforms
Let L C H = M x E? be a Willmore surface, and assume A =fi 0 on each
component of M. Wewant to describe directly the two-step Backlund trans-
form L -+ L, and compute its associated quantities (mean curvature sphere,

Hopf fields).
We state a fact about singularities that will be proved in the appendix,
see section 13.
Proposition 17. Let L be a Willmore surface in EIP', and A $ 0 on each
component of M. Then there exists a unique line bundle L C H such that on
an open dense subset of M we have:
L=kerA, andH=L(DL.
A similar assertion holds for image Q.
Weshall assume that L is immersed, and want to prove again that L is

Willmore.
Theorem 7. For the 2-step Bdcklund transform L of L we have
Q = A. (9.10)
Hence is again a Willmore surface.
Let S, 6, Q, etc. denote the operators associated with L.
Lemma11.
Proof. Since All = 0 we interpret AE f?'(Hom(H/!,, H)). On a dense open

subset of Mthen A(X) : HIL -+ H is injective for any X 54 0.
For 0 E F(L). we get
0 =d(*A)o =
d(*Ao) + *A A do = *A * do + Ado,
=0
AS * 60 + Ao = -AS(*S + SS)O.
The injectivity of A then proves the lemma.
Proof (of the theorem). Motivated by the lemma, we relate 9 to -S rather

than to S. We put
-S + B.
Then
58 9 Biicklund T ansforms of Willmore Surfaces
Q = SdS -
*dS
= Bd9 -
(Sdg + *dg)
= Bd9 -
(SdB + *dB) + (SdS + *dS)

= 4A + Bd9 -
(SdB + *dB).
The proof will be completed with the following lemma which shows that Q
-
like A -
has values in L, while the "B-terms" take values in L
Lemma12. We have
image B C (9.11)
image(*dB + SdB) c 15, (9.12)
L c kerB, (9-13)
image 0 C L. (9.14)
Proof. Recall that L is S-stable. It is of course also 9-stable, and therefore
B!, C L. (9.15)
Now L is immersive, and therefore image S HIL Thus (9. 11) will
= follow if
we can show FrBdo = 0 for 0 E F(L). But, using Lemma 11,
-kBdo =
*Sdo + irgdo =
S-Rdo + 9-kdo = SSO+ 9SO
=
-*so+ *SO =0.
Next, for X E F(H) we have
-R(*dB + SdB)X =
-k(*d(BX) + Sd(BX) -
B * dX -
SBdX)
L-valued
=
(*J + SJ)Bx
= 0. (Lemma 11)
This proves (9.12).
On the other hand, for 0 E F(L),
Fr(*dB -
SdB)o = ir (*dS -
SdS)o +Fr(*dg -
Sdg)o
=-4QO=O
= ir (*dg + gdg)o --k (Bdg)o

EF(L)
= 0.
Together with the previous equation we obtainTrdB IL =

0, and, for 0 E F(L),
9.2 Two-Step Micklund T ansforms 59
SBO -k(d(BO)) = =
*((dB),O -
Bdo) =
*dBO = 0.
But L is an immersion, and therefore BO =

0, proving (9.13).
Finally, for C- F(L),
4Q50 9d90 -
*d9O
9d9o -
do + 9 * do -
(-do + 9 * do*d9O)
+
9(dgo + 9do + *do) -
*(*do + 9dO + d9o)

=
(9 *) (d(90)
-
+ *d b)
=
-(9 *)(d(So)
- -
*do) using (9.13).
But 7r(d(SO) -
*do) =
(6S -
*6)0 = 0. So d(SO) -
MO-C 1-(L), and this

is stable under S = B -
S. Therefore QL c L. Since QL =
0, this proves
(9.14).
Taking the two-step backward transform of L, we get image Q =

image A
L. Hence L = L. We remark that the results of this section similarly apply
to the backward two-step Bdcklund. transformation L image Q. As a
corollary of (9.10) and its analog A Q we obtain
Theorem 8.
L = L = L.
10 Willmore Surfaces in SI
Let < -,
> be an indefinite hermitian inner product on EV. To be specific, we
choose
< Vi W >:--::: IT1W2 + IT2W1
Then the set of isotropic lines 1, 1

< > = 0 defines an S3 C RP1, while the
complementary 4-discs are hyperbolic 4-spaces, see Example 4. Wehave
(a db)
C
(10.1)
and the same holds for matrix representations with respect to a basis (v, w)
such that
<V'V>=O=<W'W>' <v,W>=l.
10.1 Surfaces in S3
Let L be an isotropic line bundle with mean curvature sphere S. We look

at the adjoint map M -* Z7p -+ SP* with respect to < >. Clearly S*
stabilizes Lj-, and L = L
1
implies
S*L = S*L' = L' = L.
Similarly,
1
(dS*)L =
(dS)*L C L L.
Moreover, if Qt belongs to S*, then
Qt (S*dS* -
*dS*)
(dSS -
*dS)*
4
-
I(SdS
4
+ *dS)*
-A*.

62 10 Willmore Surfaces in S3
Therefore kerQt =
(image(Qt)*)' =
(imageA)-L D L'
We proceed to show that S and S* coincide on L and HIL. By the
uniqueness of the mean curvature sphere, see Theorem 2, it then follows that
S* = S.
Let 0 E 1'(L), and write
SO =
OA, S*O Op
and
< 0,60 >=
Note that < 0, JO > makes sense, because of < 0, L >= 0. Differentiation of
0, 0 >= 0 yields
W+C0 = 0.
From 0 =< 0, So > we obtain
0 =< JO, SO>+< 0,(dS)O >+ < 0,SJO >

%-
I.,
=0
=< JO, SO > + < S*O'JO >
=< JO,O >A+ P < O,JO >
= CDA + pw.
Now we apply * using
*W =< 0, JSO >= WA, (10.2)

and obtain
0 = AOA+ pwA =
(p -
)CDA.
Weconclude A i.e. SIL = S* IL -
Now assume S I I p and S* I HIL I u. Then

HIL
= =
0 =< 50' so > + < S*0,50 >
=< S*JO' 0 > + < 0, SJO >

=& < JO'O > + <O'JO > P
&6j + WP.
But
*(.,) =< 0, JSO >=< 0, S60 >= WP
=< S*O' 90 >= XW.

10.2 Hyperbolic 2-Planes 63
Comparison with (10.2) shows p =

A, and we get
0 = & W + WA =
( -
&)WA.
It follows a = A =
p, i.e. SIHIL =
S*IHIL. This completes the assumptions
of Theorem 2, and S* = S by uniqueness.
Conversely, if S* = S and So OA, then =
A < 0,0 >=< SO,O >=< 0,SO >=< 0,,0 > A = A < 0,0 >.
Now S2 = -I implies A2 1, and therefore we get < 0, V) > = 0.
Proposition 18. An immersed holomorphic curve L in HP1 is isotropic,

i.e. a surface in S', if and only if S S*. =
10.2 Hyperbolic 2-Planes
In the half-space or model of the hyperbolic

Poincar6 space, geodesics are
euclidean circles that

orthogonally intersect the boundary. We consider the
models of hyperbolic 4-space in HP', and want to identify their totally
geodesic hyperbolic 2-planes, i.e. those 2-spheres in RP' that orthogonally
intersect the separating isotropic S3. Using the affine coordinates, from Ex-
ample 4, we consider the reflexion H -+ H, x -+ -.t at Im H S3 This =
.
preserves either of the metrics given in the examples of section 3.2. In par-
ticular, it induces an isometry of the standard Riemannian metric of RP'
which fixes S3. Given a 2-sphere S E End(EV), S2 -I, that intersects S3 =
in a point 1, we use affine coordinates, as in Example 4, with 1 =:F vH and w
such that
< V, v >=< W, W>= 0, < V, W
Then
S =
(0
N -H
-R
with N2 = R2 1, NH =
HR, and S' C Ifff is the locus of
Nx+xR=H.
If S' is invariant under the reflexion at S3, then it also is the locus of -Nd -
iR = H or
Rx + xN = T1.
According to section 3.4, the triple (H, N, R) is unique up to sign. This implies
either
(H, N, R) =
(ft, R, N) or (H, N, R) -R, -N).
By (10.1) either S* =
S, and the 2-sphere lies within the 3-sphere, or it
intersects orthogonally, and S* = -S. We summarize:
Proposition 19. A 2-sphere S E Z intersects the hyperbolic 4-spaces de-

termined by an indefinite inner product in hyperbolic 2-planes if and only if
S* = -S.
10.3 Willmore Surfaces in S' and Minimal Surfaces. in

Hyperbolic 4-Space
Let L be a connected Willmore surface in S3 C HP', where S3 is the isotropic

set of an indefinite hermitian form on H. Then its mean curvature sphere
satisfies
S* S.
Let us assume that A $ 0, and let L ker A and image Q be the 2-step
B.icklund transforms of L.
Lemma13.
L L.
Proof. First we have
I
Q*= (SdS -
*dS)* (dSS -
*dS)
4 4
I
(-SdS -
*dS) = -A. (10-3)

4
Now imap Q is S-stable, and S* = S and So =

OA imply < 0, 0 > = 0.
Therefore < L, L >= 0, and on a dense open subset of M
L = L' =
(image Q)' =
kerQ* = ker A
Lemma 14.
-S
for the mean curvature sphere 9 of L.
Proof. First L = L is obviously (-S)-stable. It is trivially invariant under A

and Q and, therefore, under d(-S) 2(*A
= -
*Q). Finally, the Q of (-S) is
I
((-S)d(-S) -
*d(-S)) =
A,
4
and this vanishes on L. The unique characterization of the mean curvature

sphere by these three properties implies 9 -S. =
10.3 Willmore Surfaces in S3 and Minimal Surfaces in Hyperbolic 4-Space 65
We now turn to the 1-step Bdcklund transform of L. If dF = 2 * A, then
d(F+F*)=2*A+2*A* =
2*A-2*Q=-dS.
(10.3)
Because S* =
S, we can choose suitable initial conditions for F such that
F+F* -S. (10.4)
We now use affine coordinates with L

[,I. I
Then the lower left entry g of
F is a Bdcklund transform of f, and (7.9) and (10.4) imply
g + 9 = H.
We want to compute the mean curvature sphere Sg. From the properties of
Bdcklund transforms we know
Nq=-R, Hg=f-f, (10.5)

see (9.3), (9.9). Likewise, 1 =
-R.. From Lemma 14 we obtain
(0 1) ( R) (1-f)=(lf
if NO
H (R (1 0 1 0 1 - ft
0)
0
-f
(1 f) (1 Hg) ( -j ) (1 -Hg) (1
9 0 -f
0 1 0 1 -fi 0 1 0 1
(1 f) (9 HgfI (1 -f)
-
01 - 0 1
This implies H and -N N- -
HH, whence
-Rg = = -N + (f -
f)H.
In particular f -
E Im H, since H = 0 on an open set would mean w 0
on that set. It follows that
(I 1) ( -g)
g -R 0
Sg =
0 f -I -N + (f -
I)H) O 1
and, because, R = N and H E R for f : M-4 Im H = R3,
(I H) (-N ; (f-f)H
g
g),
01 H
-
-
S*
g 0 1 0 1
f f
=
(Ig-H) (N+(I-f)H 0) (1H-g)
0 1 f N 0 1
(I g) ( 1-f N+(I-f)H ) (i -g)

N 0
01 0 1
-S9-
We have now shown that the mean curvature spheres of g intersect S' or-
thogonally, and therefore are hyperbolic planes. Weknow that, using affine
coordinates and a Euclidean metric, the mean curvature spheres are tangent
to g and have the same mean curvature vector as g. This property remains
under conformal changes of the ambient metric. Therefore, in the hyperbolic
metric, g has mean curvature 0, and hence is minimal. If A =- 0, then w =
0,
and the "Micklund tr 'ansform" is constant, which may be considered as a de-
generate minimal surface. In general g will be singular in the (isolated) zeros
of dg = 1w,
2
but minimal elsewhere.
Weshow the converse: Let L be an immersed holomorphic curve, minimal
in hyperbolic 4-space, i.e. with S* = -S. Then
1 1 I
A* =
(SdS + *dS)* (dSS -
*dS) (SdS + *dS) -A,

4 4 4
and therefore also
(d * A)* = -d * A.
From Proposition 15 we have
(f -f f)
I dw dw
d A
4 dw -dw f
Therefore
dw jw-, f dw =
dwf,
and hence
dw(f + f) = 0.
But f is not in S1, and therefore dw=O, i.e L is Willmore. Similarly, Propo-
sition 12 yields
*A =
(W
and A* = -A implies w = -77D. Rom S* -S we know TI =
-H, and
the backward Bdcklund transform h with dh 1w -
dH and suitable initial
conditions is in Im H = R.
To summarize
Theorem 9 [11]). Let <

(Richter > be an indefinite hermitian product
.,.
on IV. Then the

isotropic lines form an S' C HP', while the two complemen-.
tary discs inherit complete hyperbolic metrics. Let L be a Willmore surface
in S' C HP'. Then a suitable forward Bdcklund transform of L is hyper-
bolic minimal. Conversely, an immersed holomorphic curve that is hyperbolic
minimal is Willmore, and a suitable backward Bdcklund transformation is a
Willmore surface in S'. (In both cases the Bdcklund transforms may have
singularities.)
11 Spherical Willmore Surfaces in HP'
In this chapter we sketch a proof of the following theorem of Ejiri [2] and
Montiel [8], which generalizes an earlier result of Bryant [1] for Willmore
spheres in S3. See also Musso [9].
Theorem 10 (Ejiri [2], Montiel [8]). A Willmore sphere in EEP1 is a
twistor projection of a holomorphic or anti-holomorphic curve in Cp3' or, in

suitable affine coordinates, corresponds to a minimal surface in R1.
The material differs from what we have treated so far: The theorem is
global, and therefore
requires global methods of proof. These are imported
from complex function theory.
11.1 Complex Line Bundles: Degree and Holomorphicity

Let complex vector bun le. We keep the symbol J E End(H) for the
E be a
endomorphism given by multiplication with the imaginary unit i.

We denote by R the bundle where J is replaced by -J. If < > is a .,.
hermitian metric on E, then
R -+ E* = E-1, 0 -+< 0,. >
is isomorphism of complex vector bundles. Also note that for complex line
an
bundles E1, E2 the bundle Hom(Ei, E2) is again a complex line bundle.
There is a powerful integer invariant for complex line bundles E over a
compact Riemann surface: the degree. It classifies these bundles up to iso-
morphism. Here are two equivalent definitions for the degree.
-
Choose a hermitian metric < .,.
> and a compatible connection V on E.
Then < R(X, Y) 0, 0 > 0 for the curvature tensor
= R of V. Therefore
R(X, Y) (X, Y) J with a real 2-form w E fl2 (M). Define
fm
1
deg(E) := W.
27r

68 11 Spherical Willmore Surfaces in HP1
-
Choose a section 0 E V (E) with isolated zeros. Then
deg(E) := ord 0 :=
E indp 0.
O(P)=O
The index of a zero p of 0 is defined

non-vanishing section using
b a local
and holomorphic parameter z
a z (0) p. Then 0 (z) '0 (z) A (z)
for Mwith = =
for some complex function A : C C U -+ C with isolated zero at 0, and
ZY A(z)'
1 dA
indp
2-7ri
where -y is a small circle around 0.
We state fundamental properties of the degree. We have
deg(B) =
deg E-1 = -
deg E,
deg Hom(Ej, E2) deg El + deg E2.
More 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
using the first definition. The curvature tensor of a surface with Riemannian
metric < > is given by R(X,Y)
.,. K(< Y,. > X- < X,. > Y), where =
K is the Gaussian curvature. Let Z be a (local) unit vector field and < .,
>
compatible with J. Then
W(X' Y) = 1traceR
2
R(X, Y) J
K
_
(< Y, JZ >< X, Z > -

< X, JZ >< Y, Z >
2
_ < Y, Z >< X, JZ > + < X, Z >< Y, JZ >)
K(< Y, JZ >< X, Z > -
< X, JZ >< Y, Z >)
= Kdet
(<
< X'Z
Y, Z
> <
> <
X'JZ
Y, jZ
>
= K dA(X, Y).
We integrate this using Gauss-Bonnet, and find 21rX(M) =
27r(2 -
2g)
27r deg(E). For the canonical bundle
K := E-1 =
Hom(TM, C) ='fw E HomR(TM, C) I w(JX) =
iw(X)}
we therefore find
deg(K) =
2g -
2.
11.1 Complex Line Bundles: Degree and Holomorphicity 69
Definition 11. Let E be a complex vector bundle. A holomorphic structure
for E is complex linear

a map a map 0 from the sections of E into the E-
valued complex anti-linear 1-forms RE
a : r(E) -+.V(KE)
satisfying
Here 6A := !(dA+i*dA).
2
(Local) sections 0 EF(Eju) are called holomorphic,
if (90 = 0. We denote by HO(Eju) the vector space of holomorphic sections
over U.
If E is a complex line bundle with holomorphic structure, and V) E
HO(E)\10}, then the zeros of 0 are isolated and of positive index because
holomorphic maps preserve orientation. In particular, if

M is compact and
deg E < 0, then any global holomorphic section in E vanishes identically.

In theproof of the Ejiri theorem we shall apply the concepts of degree
and holornorphicity to several complex bundles obtained from quaternionic
ones. Werelate these concepts.
Definition 12. If (L, J) is a complex quaternionic line bundle, then
& := 10 E L I JO =,Oil
is a complex line bundle. We define
deg L deg EL.
Lemma15. If LI, L2 are complex quaternionic line bundles, and Ej := ELj,

then
Hom+(L1, L2) Homc(El, E2)

B BjEj
is an isomor Phism of complex vector bundles. In particular
deg Hom+(L1, L2) deg L, + deg L2
The proof is straightforward. We now discuss one example in detail.
Example 22. Weconsider an immersed holomorphic curve
LCH=MxEV
in HPI with mean curvature sphere S. The bundle K End- (H) is a complex
vector bundle, the complex structure being given by post-composition with
S. For B E r (K End- (H)) we define
70 11 Spherical Willmore Surfaces in HP1
(,9xB)(Y)V5 =
c9x(B(Y),O) -
B(OxY)o -
B(Y)o9x0,
where
Oxy:= -
V, Y1 + J1jx' YD'
2
0,0 = I(d+S*d),O,
2
i90=_I(d-S*d)0f6r'0E.V(H).
2
Direct computation shows that this is in fact a holomorphic structure, namely

that induced on
K End- (H) = K Hom+(TI, H) = K Homc(ft, H)
by 6 on TM, and the above (quaternionic) holomorphic structures 0 on H

and 0 on ft.
Lemma 16.
(d * A) (X, JX) = -
2 (Ox A) (X).
Proof. Let X be a local holomorphic vector field, i.e. [X, JX] =

0, see Re-
mark 12, and- 0 E F(H). Then
(d * A) (X, JX)o =
(-X -
A(X) -
(JX) -
SA(X) -
A([X, JX])o
1--le-I
=0
-(d(A(X),O) + *d(SA(X)O))(X)
+ A (X) do (X) + SA(X) * d b (X)

(do + *d(So)) (X) + A(X) (do -
S * do) (X).
Now
do + *d(SO) =
(c9 + 0 + A+ Q)o + *(,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
do-S*do= (o9+5+A+Q)0-S*(o9+O+A+Q)0
=
(a + 0 + A+ Q)o -
S(SO -
SO + SA -
SQ)0
=
(0 + 0 + A+ Q),0 -
(-,9 + 6 -
A+ Q)o
=
2(0 + A)O.
11.2 Spherical Willmore Surfaces 71
Therefore
(d * A) (X, JX),o = -20x (A(X)O) -
2A(X)2,0 + 2A(X)L9xO + 2A(X)2,0

-2(0x(A(X)0) -
A(X)c9xO)
_2(6xA)(X)0-
Now assume Willmore,
that L is and therefore d * A = 0. This implies
6A =
0, and A is holomorphic:
A E Ho (K End- (H)) = Ho (K Hom+(TI, H)).
As a consequence, see Lemma23, either A _= 0, or the zeros of A are isolated,

and there exists a line bundle L C H such that ker A away from L = the
zeros of A. For local 0 E F(L) and holomorphic Y E HI(TM) we have
6A (Y)o =
O(A(Y)O) -
A(Y)&O.
=0 =0
Therefore L is 49, like L is

invariant under invariant under 0, see Re-
mark 6. Asabove, get holomorphic structure
we a on the complex line bundle
K Hom+(TIlL, L) and A defines a holomorphic section of this bundle:
A E Ho (K Hom+(RIL, L)).
11.2 Spherical Willmore Surfaces
We turn to the .
Proof (of Theorem 10). If A =- 0 or Q E 0, then L is a twistor projection by

Theorem 5.
Otherwise we have the line bundle L, similarly
and a line bundle L that
coincides with the image of Q almost everywhere.
Proposition 20. We have the following holomorphic sections of complex

holomorphic line bundles:
A E Ho (K Hom+ L)), Q E Ho (K Hom+(HIL, LA

JLEHO(KHom+(L,H/L)), AQEHO(K Hom+(HIL, L)) 2
and if AQ = 0 then J1 E Ho (K Hom+(L, HIL))
We proved the statement about A. We give the (similar) proofs of the

others appendix.
in the
The degree formula then yields
72 11 Spherical Willmore Surfaces in IFffP1
ord JL =
deg K -
deg L + deg HIL

ord(AQ) = 2 deg K -
deg HIL + deg L
3 deg K -
ord 6L
6(g -
1) -
ordJL.
For M= S2, i.e. g =

0, we get ord(AQ) < 0, whence AQ 0. Then
and
ordA =
deg K+ degH/L + degL
ordQ =
degK -
degH/L -
degL
ord 6.L =
deg K+ deg L -
deg HIL.
Addition yields
ordJ.L + ordQ + ordA = 3 degK -
deg HIL + degL

= 4 deg K -
ord JL = -8 -
ord JL.
It follows that ord 51 < 0, i.e. 6i 0, and L is d-stable, hence constant in

H = Mx IBF. From AS = -SA L. Therefore all
0 we conclude A
mean curvature spheres of L pass through the fixed point Choosing affine
coordinates with L oo, all mean cur vature spheres are affine
=
planes, and L
corresponds to a minimal surface in W.
12 Darboux tranforms
Bdcklund transforms provided a mean to construct new Willmore surfaces

out of a given one by solving linear equations. Darboux transforms provide
another method for such construction, based on the solution of a Riccati
equation. For isothermic surfaces it is described in [6].

After an introductory remark on Riccati equations, we first describe Dar-
boux transforms for a special case of [6], namely for constant mean curvature
surfaces in R, because it displays a striking similarity with the Willmore

case treated thereafter.
As with the Bdcklund transforms the theory -
in the Willmore case -
is again local. We only have a local existence of a solution to the Riccati
initial value problem, and moreover require this solution to be invertible in

the algebra End(IV).
12.1 Riccati equations
We consider Riccati type partial differential equations in an algebra, which

below will be H or End(EP).
Lemma17. Let A be an associative unitary algebra over the reals, and M
a manifold. Let a,# E S?1 (M, A) with
da = 0 =
d,
a A P = 0 =
PAa.
Then for any p E R\ 10}, po E M and To E A the Riccati initial value problem
dT =
pTa T -
0, T(po) = To (12.1)
has a unique solution T on a connected neighborhood of po.

Moreover, if S: M- A with
S2 =
_1' Sa+aS=O, dS=a-0,
and

74 12 Darboux tranforms
(T -
S)2(po) =
p-1
then
(T -
S)2 = P-1 (12.2)
everywhere, and T'S = S2T.
Proof. The integrability condition for (12.1)
0 =
pdT A aT + pTdaT -
pTa A dT -
d#
=
p(pTaT -
0) A a T+ pTdaT -
pTa A (pTaT -
0) -
dfl
=
-pp A aT + pTdaT + pTa A 0 -
do
is obviously satisfied. Now, if T is a solution, and S as above, then we define
X:= p(T -
S)2 + P _
1.
Then X satisfies a linear first order differential equation
p-'dX =
(dT -
dS)(T -
S) + (T -
S)(dT -
dS)
=
(pTaT -
a) (T -
S) + (T -
S) (pTaT -
a)
=
Ta(pT2 -
pTS -
1) + (pT 2-
pST -
1)aT + aS + Sa
=0
= TaX + XaT.
Hence X(po) = 0 implies X = 0. The last equation of the lemma follows from
T 2S_ ST2 =
(T _
S)2S -
S(T _
S)2
together with (12.2).
12.2 Constant mean curvature surfaces in R'
Let f : M-+ Im H be a conformal immersion:
*df =
Ndf =
-dfN, N2 = _1.
dN is a "'tangential"' 1-form: it anticommutes with N. We decompose it

into the K- and ff-part with respect to the complex structure given by left
multiplication by N to obtain
dN = Hdf + w. (12.3)
Since that is also the decomposition of the shape operator into "trace" and
traceless part,the function H : M --- R is the mean curvature, and *w
-Nw. Then
12.2 Constant mean curvature surfaces in R3 75
1(dN
2
+ N* dN).
Note that (12.3) resembles the formula dS 2 * Q = -

2 * A.
Now -dN is the shape operator of f, and (12.3) gives its decomposition
into the "
trace" and the traceless part. Therefore H M -4 R is the mean
curvature. Differentiating (12.3) we get
O=dHAdf +dw.
Hence H is constant if and only if dw =

0, resembling d * Q 0.
Wesee that the theory of constant mean curvature (=cmc) surfaces in R3
parallels that of Willmore surfaces in HP1.

Wenow assume H to be constant 54 0. Then
g:=
IN
H
satisfies
1
dg = df dN
jy
-
dN =
H(df -
dg),
*dg =
-Ndg =
dg N,
and
df Adg=O=dgAdf
by type. The map g is an immersion of constant mean curvature H away

from the umbilics of f, i.e. away from w = 0. It is called the parallel constant
mean curvature surface of f.

For simplicity we restrict ourselves to the case
H = -1.
(The general case can be reduced to this using the homothety f -+ Pf, H
H
P
, g -+ tig with [t =
-H.)
We put A := End(p, a =
dg, 0 df =
.
These match the assumptions of
lemma 17. Therefore, for any jo 0 O,pO E M and To E ImH\10} the initial
value problem
dT =
pTdgT -
df, T(po) = To
(locally) has unique solution T which we assume to have no zeros. T stays

,
in Im A because T satisfies the same equation up to a minus sign.

We put
fO =
f +T.
Then
*df *(df + dT) =

pT * dgT =
-pTNdgT =
-TNT-'pTdgT
-TNT-' (df + dT) =
-TNT-ldf 0.
shows that 0 is M:=

This f conformal with Nf o = -TNT-'. Moreover, f
is an immersion if and only if g is an immersion. if g is immersive.
Under what conditions does f 0 again have constant mean curvature? We
compute HO := Hf o, using
T2 =
-IT12 , TN+NT=TN+TN=-2<T,N>,
and
dN A df =
Hdf A df -
Hdg A df =
Hdf A df.
Wefind
HOdf 0 A df 0 = dNO A df
=
-d(TNT-1) A df 0
=
-(dTNT-1 + TdNT-1 TNT-1dTT-1)
-
A pTdgT
=
-(dTN + TdN -
TNT-1dT) A pdgT
=
(-(pTdgT -
df)N -
Tdg + TNT-' (pTdgT -
df)) A pdgT
=
-p(Tdg(TN + NT) + p-'Tdg)) A pdgT
2 < T,N > -p-I
pTdgT A pTdgT
IT12
2 < T,N > -jo-1 0 A
df df
IT12
Hence we proved
Lemma 18.
HO =
2 < T,N > -p-I
IT12
Next we show
Lemma 19. If HO is constant, then H = -1.
Proof. We differentiate 0 =
HOIT12 + 2 < T, N> -p-1:
12.2 Constant mean curvature surfaces in 77
0 = HO < dT,T > + < dT,N > + < T,dN >
=
HO(< pTdgT,T > -
< df,T >)+ < pTdgT,N > -
< T,df > + < T,dg >
=
HO(-IT12p < T,dg >) - HO < df,T > + < pTdgT,N >
-
< T,df > + < T,dg >
=
-(HOIT12P _
1) < T,dg -(HO + 1) < T,df > + < pTdgT,N

> >
=
2p < T,N >< T,dg > -(HO + 1) < T,df > + < pTdgT,N >,
=
-(HO + 1) < T, df >
P
+ ((TN + NT) (Tdg + dgT) -
(TdgTN + NTdgT))
2
=
-(HO + 1) < T,df
+ (Tdg(TN + NT) + (TN + NT) dgT -
(TdgTN + NTdgT))
2
=
-(HO + 1) < T, df >
+ (TdgTN + TdgNT + TNdgT +NTdgT -
(TdgTN + NTdgT))
2 %
-V
=0
=
-(H# + 1) < T, df >.
If HO 1, we are done. Otherwise < T, df >= 0, i.e. T =

jLN, and
dT =
dpN + pdN djzN -
ydf + pw
dT =
pTdgT -
df PP2 NdgN -
df =
ptt2W -
d'
Now df and w are tangential, and comparison of the above two equations
gives
dl-t =
0,
(I _
tl)df (_tl + P,12)W,

dl-t = 0 and therefore
But then f g is the parallel constant mean curvature surface of f which

has HO = -1.
As a consequence of the preceeding two results we obtain
Lemma20. HO is constant, if and only if
(T -
N)2 =
P-1 (12.4)
Proof. Weknow that HO ist constant, if and only if it equals -1, and this is
equivalent with
IT 12 -
2 < T,N > +p-1 = 0.
But
(T -
N)2 =
-IT -
N12 =
-(IT12 -
2 < T,N > +1)
=
-(ITj2 -
2 < T,N > +p-1) + p-1 -
I.
Now recall from lemma 17 that (12.4) holds everywhere, if it holds in a single
point. Therefore IT- S12 = 1 -
p-1, and T is bounded with no zeros. Hence it

can be globally defined if Mis simpli connected. This leads us to the following
Definition 13. Let f : M-4 Im H be a conformal immersion with constant
mean curvature H= -1, and immersed parallel cmc surface g f = + N. Let
p E R\101, To E ImH\101, po E M,
and assume
(To -
N(po))2 =
P-1 _
1. (12.5)
Let T be the unique solution of the Riccati initial value problem
dT=pTdgT-df, T(po)=To.
Then
fO:=f+T
is called a Darboux transform of f.
Remark 11. 1. If H is constant 0 0, -1, then (12.5) in the above definition

should be replaced by
H
(HTo + N(po ))2 = 1.
P
It turns out that f + T has

again constant mean curvature H.
2. From (12.5) we see that
a given p 54 0 there is an S2 of initial
for To
Hence there is a 3-parameter family of Darboux transforms.
We summarize the previous results:
Theorem 11. The Darboux transforms of surfaces with constant mean cur-
vature H in R' have constant mean curvature H.

12.3 Darboux transforms of Willmore surfaces 79
12.3 Darboux transforms of Willmore surfaces
Let L C H = Mx EV be a Willmore surface in HP1 with mean curvature

sphere S, and dS =
2(*A -
*Q). Since d * A = d Q = 0 we can define

End(H2)-valued maps F, G, locally on Mby
dF=2*A, G=F+S.
Then
dG = 2 * Q,
dS = dG -
dF,
dG A dF = 0 = dF A dG.
Hence the integrability conditions for the Riccati equation in A End(fffl,

with a =
dG, # =dF are satisfied.
As in the cmc case we find for any
p E R\10}, To E GL(2, R, po E M
a unique (local) solution T of the Riccati initial value problem
dT =
pTdG T -
dF, T (po) =
To,
which we may assume to be invertible. As above let us assume that
(T 0 _
S( P0))2 =
(P
Then
(T -
S)2 =
(P-1
everywhere by lemma 17, and we call
LO := T-1L
a Darboux transform of L.
Our aim now is to show that V is again Willmore. Westart by computing
its mean curvature sphere.
Lemma21. The mean curvature sphere of V is given by
S0:=T-1ST=TST-1,
and the corresponding Hopf fields are
2*AO:=P-'T-'dFT-1, 2*QO:=pTdGT.
Proof. First note that the derivative 60 E 0'(Hom(LO, HILO)) of LO is given

by
50 = T-16T.
Therefore LO is immersed and
*50 = T-1 * 6T = T-'SJT = T-'STT-16T = S060.
A similar computation yields *60 = 60 SO -

Due to the definition of SO and LO
we obviously have
SOLO = LO.
Moreover,
T-1ST = T-1ST2T-1 = T-'T2ST--1 = TST-1.
Now
dSO =,d(TST-1)
= dTST-1 + TdST-1 -
TST-'dTT-1
=
(pTdGT -
dF)ST-1 + TdST-1 -
TST-l(pTdGT -
dF)T-1
=
T((pdGT -
T-'dF)S + dS -
S(pdGT -
T-'dF))T-1
=
T(pdG(TS p-II) + (T-1S
+ ST + + ST-1 -
I)dF)T-1
=
T(pdGT 2-
jo-1T -2
dF)T-1
=
TpdG T p-'T-'dF T-1-
= 2(*QO -
*AO),
which is the decomposition of dSO into type:
*TdG T = T * (2 * Q)T =
-TS(2 * Q)T = -TST-'TdG T = -SOTdGT,
and similarly for F.

Finally,
QOILO =
0)
and AOH2 C LO, whence
dSOLO C LO.
This proves that SO is the mean curvature sphere of LO.

12.3 Darboux transforms of Willmore surfaces 81
Theorem 12. The Darboux transforms.of an immersed Willmore surface in

HP' are again Willmore surfaces.
Proof.
-2p-ld * QO =
d(TdGT) = dT A dGT -
TdG A dT
=
(pTdGT -
dF) A dGT -
TdG A (pTdGT -
dF)
=
p(TdGT A dG -
TdG A TdGT) = 0.
13 Appendix
13.1 The bundle
Lemma22. If L is is an immersed holoMorphic curve in HP1, then
AIL -= 0 4=* A = 0.
Proof, Let AIL = 0. Since QJL =

0, we find for 0 E F(L)
0 =
d(*Qo) =
(d * Q),O -
*Q A dO =
(d * Q)O =
(d * A) V).
(5.2)
Note that *Q A do =
*Q A JO = 0 by type, where Q : H -+ HIL. Since
AIL =
0, we now obtain similarly
0 =
d(*Ao) =
(d * A)O -
* AA do AA &0 A* JO -
AJO =
-2AJO.
=0
But L is an immersion. Therefore AIL 0 = AJ implies A = 0. The converse
is obvious.
Lemma23. Given a holomorphic section T E HO(Hom(V, W)), where V, W

are holomorphic complex vector bundles, there exist holomorphic subbundles
VO C V, fV C W
such that Vo = ker T and T7 =

image T away from a discrete subset.
Proof. Let r := maxf rank Tp I p E MI and G := fp I rank Tp r}. This is =
an open subset of M. Let po be a boundary point of G, an let 01,

, on
be . . .
holomorphic sections of V on a neighborhood U of po. By a change of indices

we may assume that To, A ...
A To, $0. But this is a holomorphic section
of the holomorphic bundle AIWIU, and hence has isolated zeros, because
dimc M = 1. We assume that po is its only zero within U. Moreover, there
exist k E N, a holomorphic coordinate z centered at po, and a holomorphic
section a E HO(ArWlu) such that
To, A ...
A To, = zku.

84 13 Appendix
Off po the section a is decomposable, and since the Grassmannian G,(W) is

closed Ar(W), it defines a section
in of G,(W), i.e. an r-dimensional subbun-
dle of WJUextending imageTIU\p.. The statement about the kernel follows
easily using the fact that ker T is the annihilator of image T* : W* -+ V*.
Proposition 21. Let L be a (connected) lVillmore surface in HP', and A ;t

0. Then there exists a unique line bundle L C H such that on an open dense
subset of M we have:
L = kerA and H = L (D L.
Proof. A E F(KEnd-(H)) is a holomorphic section by Example 22. By

Lemma23 there exists a line bundle L such that L = ker A off a discrete set.
Assume now that H,, =,4 Lp E) Lp for all p in an open non-empty set U C M.
Then L = L, and AIL 0 on U.
= But then Alu = 0 by Lemma 22. This is a
contradiction, because the zeros of A are isolated.
13.2 Holomorphicity and the Ejiri/Montiel theorem
In this section L denotes an immersed holomorphic curve in UP'.
Remark 12 (Holomorphic Vector Fields). The tangent bundle of a Riemann

surface viewed as complex line bundle carries a holomorphic structure:
I
oxy =
([X, Y] + J[JX' Y]).
2
Note that this is tensorial in X. The vanishing of the Nijenhuis tensor implies
5J = 0. A vector field Y is called holomorphic if OY 0. This is equivalent =
with 5yY = 0 =
OjyY, but either of these conditions simply says
[Y' JY] = 0.
Any constant vector field in C is therefore holomorphic, and a given tangent

vector to a Riemann surface can always be extended to a holomorphic vector
field.
Proposition 22. Let L be a Willmore surface in HP'. Wehave the following

holomorphic sections of complex holomorphic line bundles:
A Ho (K Hom+(RIL, L)), Q (=- Ho (K Hom+(HIL, L)),

JL Ho (K Hom+(L, HIL)), AQ E H'(K 2
Hom+(HIL, L)),
7;
Ho (K Hom+(L, HIL)).
~
if AQ
-
and = 0 then JL E
For the proof we need

13.2 Holomorphicity and the Ejiri/MOntiel theorem 85
Lemma24. The curvature tensor of the connection a +0 on H is given by
R'9+'9 =
-(A AA+ Q A Q),
and for a holomorphic vector field Z we have
R'9+'5(Z, JZ) =
2S(Ozaz -
OZOZ). (13.2)
Proof. In general, if V and V+ w are two connections, then
RV =RV+d7w+wAw.
We apply this to 0 + 0 = d -
(A + Q) and use Lemma4:
Ra+'9 R
d
-
d(A + Q) + (A + Q) A (A + Q)
-2(A AA+ QA Q) + (A AA+ Q A Q)
-(A AA+ Q A Q).
Equation (13.2) follows from
Ro+'9 (Z, JZ) =

(az 5z) (aiz + Oiz)
+ -
(Oiz + Oiz) (az + 6z) .
= S 09Z + 6z) (az 6z) - -

S (az -
6z) (az + 6z)

=
2S(-o9z6z + Ozaz),
becauseOZ2 = 0 = a2Z*
Proof (Proof
of the proposition).
The holomorphicity, of A was shown in example 22, and that of Q can be
shown in complete analogy.
(H, S) is a holomorphic complex quaternionic vector bundle, and L is
a holomorphic subbundle, see Remark 6. Therefore L and HIL are holo-
morphic complex quaternionic line bundles, and the complex line bundle
K Hom+(L, EIL) inherits a holomorphic structure. Then, for local holomor-
phic sections 0 in L and Z in TM,
(49Z6L)(Z)'O =
Mh(Z)O) JL(aZZ)O -
6L(Z)(09ZO)
= aZ (Z) 19Z (7rL (10 (Z))
=
7rD9Z(dO(Z)) =
7TL49Z(19ZV))-
By (13.1) and (13.2) we have
OzOzO = azOzO --IS R9+9(Z, JZ)O,

2 ,
I.-
=0 EL
hence
86 13 Appendix
(OZ JL) (Z) = 0 -
Then also
(,9jzjL)(Z) =
S(,9zjL)(Z) =
0,
and therefore 06L :-- 0.

To prove the holomorphicity of AQ E r(K 2Hom(H/L, L)), we first note
that
2
K Hom(H/L, L) = Homc(TM, Homr (TM, Hom+(HIL, L)))
carries a natural holomorphic structure. The rest follows from the holomor-
phicity of A, Q, and the product rule.
Finally we interpret 6.E as a section in K HoM+ (L, Note that the
holomorphic structure on TI is given by a. From the holomorphicity of A we
find, for 0 E 1'(L),
0 =
(aA)O =
a(AO) + MO.
=0
This shows that L is a-invariant. Moreover, it is obviously invariant under A

and, as a consequence of AQ =
0, also under Q. RomLemma 24 it follows
that 1, is invariant under R9+5, and that for a local holomorphic vector field
Z and a local holomorphic section 0 of L,
I -
SRO+'9 (Z, JZ),O.

- -
azazO = Oz azO + -
2, V
=0
EL
Then
(OZ h) (Z) 0 = 19 (h (Z) 0) -
h (OZ Z) 0 h (Z) az 0 -
09Z (JI (Z)O) =

az(ir_LdO(Z)) -7rIaz(dO(Z)) =
=
7rLaZOZO = 0.
References
1. Bryant, Robert. A duality theorem for Willmore surfaces. J. Differential Geom.

20, 23-53 (1984)
2. Ejiri, Norio. Willmore Surfaces with a Duality in SN (1). Proc. Lond. Math.
Soc., III Ser. 57, No.2, 383-416 (1988)
3. Ferus, Dirk. Conformal Geometry of Surfaces in S4 and Quaternions. In: Sum-
mer School on Differential Geometry, Coimbra 3/7 September 1999, Proceed-
ings, Ed.: A.M. d'Azevedo Breda et al.
4. Ferus, Dirk; Leschke, Katrin; Pedit, Franz; Pinkall, Ulrich. Quaternionic Phicker
Formula and Dirac Eigenvalue Estimates. To appear in Inventiones Mathemat-
icae
5. Friedrich, Thomas. On Superminimal Surfaces. Archivum math. 33, 41-56
(1997)
6. Hertrich-Jeromin, Udo; Pedit, Franz. Remarks on Darboux Transforms of
Isothermic Surfaces. Doc. Math. J. 2, 313 333 (1997).
-
(www.mathematik.uni-bielefeld.de/documenta/vol-02/vol-02.html)
7. Kulkarni, Ravi; Pinkall, Ulrich (Eds.). Conformal Geometry. Vieweg, Braun-
schweig 1988
8. Montiel, Sebasti6n. Spherial Willmore Surfaces in the Fo,4r-Sphere. Preprint
1998
9. Musso, Emilio. Willmore surfaces in the four-sphere. Ann. Global Anal. Geom.
13, 21-41 (1995)
10. Pedit, Franz; Pinkall, Ulrich. Quaternionic analysis on Riemann surfaces and
differential geometry. Doc. Math. J. DMV, Extra Volume ICM 1998, Vol. II,
389-400.
(www.mathematik.uni-bielefeld.de/documenta/xvol-icm/05/05.html)
11. Richter, J8rg. Conformal Maps of a Riemannian Surface into the Space of
Quaternions. Dissertation, Berlin 1997
12. Rigoli, Marco. The conformal Gauss map of Submanifolds of the Moebius Space.
Ann. Global Anal. Geom 5, No.2, 97-116 (1987)
13. Tenenblatt, Keti. Transformations of Manifolds and Applications to Differential
Equations. Chapman& Hall/CRC Press 1998

Index
affine coordinates 9 line subbundle 16
Bdcklund transform 53,'55-57 mean curvature sphere 27,44

mean curvature vector 40,42
complex quaternionic bundle 18
minimal surface 66
conformal 7, 8
conformal curve 19
normal curvature 40
conformal Gauss map 29
normal vector 6
constant mean curvature surface 75
Darboux transform 78, 81 projective space 9
degree 67
derivative of L 17 quaternionic vector bundle 15
differential Jf 11
dual bundle 21 Riccati equation 73
energy functional 31 second fundamental form 39

suPer-conformal 50
Gaussian curvature 40
harmonic map 31 tangent space of projective space 10
holomorphic curve 19,50 tautological bundle 15

twistor projection 47, 50
holomorphic quaternionic bundles 20
holomorphic section 84 two-spheres in S' 13
type argument 28
holomorphic structure 69
Hopf fields 27
Willmore cylinder 46
hyperbolic 2-planes 64
hyperbolic minimal 66 Willmore functional 35
Willmore surface 36, 55, 81
isotropic curve 63 -
constrained 36

Francis E. Burstall, Dirk Ferus, Katrin Leschke, Franz Pedit, Ulrich Pinkall - Conformal Geometry of Surfaces in Quaternions-Springer (2002)

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3

Francis E. Burstall Franz Pedit

Dept. of Mathematical Sciences Dept. of Mathematics

CatalogIng-in-Publication Data applied for.

Conformal geometry of surfaces in S4 and quaternions / E E. Burstall ....

Mathematics Subject Classification (2000): 53C42, 53A30

Typesetting: Camera-ready TEX output by the authors

This is the firstcomprehensive introduction to the authors' recent attempts

Summer School on Differential Geometry at Coimbra in

quaternionic theory with notions in classical submanifold theory. The rest

R3 and that of Willmore surfaces in EEP1, and use it to construct Darboux

Bath/Berlin, Francis Burstall, Dirk Ferus, Katrin Leschke,

2 Linear Algebra over the Quaternions ..............

3 Projective Spaces .........................................

4 Vector Bundles ...........................................

5 The Mean Curvature Sphere .............................

6 Willmore Surfaces ........................................ 31

7 Metric and Affine Conformal Geometry ..................

8 Twistor Project"ions ...................................... 47

9 Bhcklund Transforms of Willmore Surfaces ...............

10 Willmore Surfaces in S3 .................................. 61

11 Spherical Willmore Surfaces in HPI ...................... 67

12 Darboux tranforms ....................................... 73

References ................................... .................

1.1 The Quaternions

The Hamiltonian quaternions H are the unitary R-algebra generated by the

symbols i, j, k with the relations

The multiplication is associative but obviously not commutative, and each

ciative and have no zero-divisors. For the element

Ima:= ali + a2i + a3k.

imaginary unit, any.purely

F. E. Burstall et al.: LNM 1772, pp. 1 - 4, 2002

Occasionally we shall need the Euclidean inner product on R4 which can

< a, b >R= Re(ab) =

ab=axb- <a,b>R- (1-3)

Lemma1. For a, b G H we have

vanishes if and only if a is real or purely imaginary. Together with (1.2) we

1.2 The Group S'

The set of unit quaternions

interpret it as the group of linear maps x -+ px of H preserving the hermitian

< a, b >:= 51b.

This group is called the symplectic group Sp(l).

S3 X H __ H, (/-t, a) 1-4 pap-1.

complement R' = Im H. We get a map, in fact a representation,

,X : '53 -+ SO(3), y -+ y ... P-1 II.H-

Let us compute the differential of 7r. For p E S3 and v E Tt'S3 we

Now y-1v commutes with all a E ImH if and only if v =

and only if M 1, this covering'is

SO(3)- S31111 Rp3

If we identify H = C G) C?, we can add yet another isomorphism:

Because of yopo + /-tlpj. =

2.1 Linear Maps, Complex Quaternionic Vector Spaces

two options for the multiplication by scalars. We choose quaternion vector

spaces to be right vector spaces, i.e. vectors are multiplied by quaternions

V x H -+ V, (v, A) F-+ vA.

then for any p E H\ f 01 we find

2-sphere of "associated eigenvalues" (see Section 1.2). This is related to the

F. E. Burstall et al.: LNM 1772, pp. 5 - 8, 2002

Remark 1. On a real 2-dimensional vector space U each complex structure

Lemma2 (Fundamental lemma).