Proc. Summer School on Diff. Geometry Dep. de Matemática, Universidade de Coimbra September 1999, 19-38.

Conformal Geometry of Surfaces in S^{4} and Quaternions

D. Ferus

Fachbereich Mathematik, Technische Universität Berlin Str. des 17.Juni 135, D-10623 Berlin e-mail: ferus@math.tu-berlin.de

Quaternionic methods recently proved to be a powerful tool in conformal surface theory. A first account of this was presented in the ICM 98 article [5], and the theory was developed further by the authors of [1] in an informal seminar at Technische Universität Berlin since then ^{1}. I presented this research to the Summer School on Differential Geometry at Coimbra 1999 in a series of lectures that gave an elementary introduction to the new concepts, but also included several recent results with complete proofs. The present article contains an extended abstract of my lectures, while the details will appear in [1].

Contents

20 D. Ferus: Conformal Geometry of Surfaces in S^{4} and Quaternions

1 Linear Algebra over the Quaternions 20 2 Projective Spaces 21 3 Vector Bundles 23 4 The Mean Curvature Sphere 26 5 Willmore Surfaces 28 6 Metric and Affine Conformal Geometry 29 7 Twistor Projections 31 8 Bäcklund Transforms of Willmore Surfaces 32 9 Willmore Surfaces in S^{3} 34 10 Spherical Willmore Surfaces in ℍP^{1} 35 1. Linear Algebra over the Quaternions

1Research supported by Sfb288 at TU Berlin.

We consider the skew-field ℍ of the Hamiltonian quaternions in the usual representation

a = a_{0} + a_{1}i + a_{2}j + a_{3}k, a_{l} ∈ ℝ,  (1.1)

with imaginary units i, j, k. We define

ā := a_{0} − a_{1}i − a_{2}j − a_{3}k,

Re a := a_{0}, Im a := a_{1}i + a_{2}j + a_{3}k,

< a, b >_{ℝ}= Re(āb) = Re(ab̄) = \frac{1}{2}(āb + b̄a).

We shall identify the real vector space ℍ in the obvious way with ℝ^{4}, and the subspace of purely imaginary quaternions with ℝ^{3}:

ℝ^{3} = Im ℍ.

The quaternion multiplication incorporates both the usual vector and scalar products on ℝ^{3}. For a, b ∈ Im ℍ = ℝ^{3} we have

ab = a × b − < a, b >_{ℝ} .  (1.2)

In particular, the usual two-sphere is given by

S^{2} ⊂ ℝ^{3} = Im ℍ = {a ∈ ℍ | a^{2} = −1}.

We choose quaternion vector spaces to be right vector spaces, i.e. vectors are multiplied by quaternions from the right. We shall often have an additional complex structure on V, acting from the left, and hence commuting with the quaternionic structure. In other words, we consider a fixed J ∈ End(V) such that J^{2} = −I. In this case we call (V, J) a complex quaternionic (bi-)vector space. If (V, J) and (W, J) are such spaces, then the quaternionic linear maps from V to W split as a direct sum of the real vector spaces of complex linear and anti-linear homomorphisms:

Hom(V, W) = Hom_{+}(V, W) ⊕ Hom_{−}(V, W).

Lemma 1. (Fundamental lemma) If N, R ∈ ℍ with N^{2} = −1 = R^{2}, then σ(x) := NxR defines a self-adjoint endomorphism of ℍ = ℝ^{4}, and its ±1-eigenspaces U, U^{⊥} are real 2planes. Conversely, each real 2-plane in ℍ determines a pair (N, R) of imaginary units unique up to sign. An orientation of U can be used to fix that sign.

If U ⊂ Im ℍ, then N = R, and this is perpendicular to U. We, therefore, call N and R the left and right normal of U though, in general, they are not at all orthogonal to U. Conformal maps of Euclidean spaces are maps with differential preserving the metric up to a factor. In the case of maps from ℝ^{2} = ℂ into itself, this can be characterized in terms of the complex structure J : z ↦ iz alone, not referring to the metric. In fact, f is conformal, if and only if it satisfies the Cauchy-Riemann equation

∗df := df ∘ J = ±J ∘ df,

where the sign depends on the orientation behaviour of df. We generalize this to maps into the quaternions in the following fundamental Definition. Let M be a Riemann surface, i.e. a 2-dimensional manifold endowed with a complex structure J : T M → T M, J^{2} = −I. A map f : M → ℍ = ℝ^{4} is called conformal, if there exist N, R : M → ℍ such that, with ∗df := df ∘ J,

N^{2} = −1 = R^{2}  (1.3)
∗df = Ndf = −dfR.  (1.4)

If f is an immersion then (1.3) follows from (1.4), and N and R are unique, called the left and right normal vector of f.

Equation (1.4) is a Cauchy-Riemann equation with a “variable complex structure” on the target space side. In this sense conformal maps into ℍ are a generalization of complex holomorphic maps. For an immersion f the existence of N : M → ℍ such that ∗df = Ndf already implies that the immersion f : M → ℍ is conformal. Similarly for R. If f : M → Im ℍ = ℝ^{3} is an immersion then N = R is “the classical” unit normal vector of f. But for general f : M → ℍ, the vectors N and R are not orthogonal to df(T M).

2. Projective Spaces

In complex function theory the Riemann sphere ℂP^{1} is more convenient as a target space for holomorphic functions than the complex plane. Similarly, the natural target space for conformal immersions is ℍP^{1}, rather than ℍ. We therefore give a description of the quaternionic projective space. The quaternionic projective space ℍP^{n} is defined, similar to its real and complex cousins, as the set of quaternionic lines in ℍ^{n+1}. We have the (continuous) canonical projection

π : ℍ^{n+1}\{0} → ℍP^{n}, x ↦ π(x) = [x] = xℍ.

The manifold structure of ℍP^{n} is defined as follows: For any linear form β ≠ 0 on ℍ^{n+1}

u : π(x) ↦ x < β, x >^{−1}

is well-defined and maps the open set {π(x) | < β, x >≠ 0} onto the affine hyperplane β = 1, which is isomorphic to ℍ^{n}. Coordinates of this type are called affine coordinates for ℍP^{n}. They define a (real-analytic) atlas for ℍP^{n}. The set

{π(x) | < β, x >= 0}

is called the hyperplane at infinity.

Example. In the special case n = 1, the hyperplane at infinity is a single point: ℍP^{1} is the one-point compactification of ℝ^{4}, hence “the” 4-sphere:

ℍP^{1} = S^{4}.

The projective line is a perfect model for the conformal structure of S^{4}. On the other hand, for instance the antipodal map is natural on the usual 4-sphere, but not on ℍP^{1} – unless we introduce additional structure, like a metric.

The tangent space of ℍP^{n} admits a well-defined isomorphism

T_{l}ℍP^{n} ≅ Hom(l, ℍ^{n+1}/l)  (2.1)

using the following

Proposition 1. Let f̃: M → ℍ^{n+1}\{0} and f = πf̃: M → ℍP^{n}. Let p ∈ M, l := f(p), v ∈ T_{p}M. Then

d_{p}f : T_{p}M → T_{f(p)}ℍP^{n} = Hom(f(p), ℍ^{n+1}/f(p))

is given by d_{p}f(v)(f̃(p)λ) = π_{l}(d_{p}f̃(v)λ). We denote the differential in this interpretation by δf:

δf(v)(f̃) = df̃(v) mod f.  (2.2)

Given a non-degenerate quaternionic hermitian inner product < ., . > on ℍ^{n+1}, we define a (possibly degenerate Pseudo-) Riemannian metric on ℍP^{n} as follows: For x ∈ ℍ^{n+1} with < x, x >≠ 0 and v, w ∈ (xℍ)^{⊥}, we put

< d_{x}π(v), d_{x}π(w) >= \frac{1}{< x, x >} Re < v, w > .

Example 1. For < v, w >= ∑ v̄_{k}w_{k} we obtain the standard Riemannian metric on ℍP^{n}. (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. Comparison with the stereographic projection shows that on ℍP^{1} it induces a metric of constant curvature 4.

Example 2. If we consider an indefinite hermitian metric on ℍ^{n+1}, then the above construction of a metric on ℍP^{n} fails for isotropic lines (< l, l >= 0), but these points are scarce. We consider the case n = 1, and the hermitian inner product < v, w >= v̄_{1}w_{2} + v̄_{2}w_{1}. Isotropic

lines are characterized in affine coordinates h : x ↦ (\begin{matrix}x \\ 1 \end{matrix}) by x̄+x = 0, i.e. by x ∈ Im ℍ = ℝ^{3}.

Therefore, the set of isotropic points is a 3-sphere S^{3} ⊂ S^{4}, and its complement consists of two open discs or – in affine coordinates – two open half-spaces. The induced metric is – up to a constant factor – the standard hyperbolic metric on these half-spaces.

Let

Z = {S ∈ End(ℍ^{2}) | S^{2} = −I}

and, for S ∈ Z, define S′ := {l ∈ ℍP^{1} | Sl = l}. Using affine coordinates with ∞ ∈ S′ and Lemma 1 one shows that

Z is the set of oriented 2-spheres in S^{4} = ℍP^{1}.

3. Vector Bundles

A quaternionic vector bundle π : V → M of rank n over a smooth manifold M is 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 3. The points of the projective space ℍP^{n} are the 1-dimensional subspaces of ℍ^{n+1}. The tautological bundle

π : Σ → ℍP^{n}

is the line bundle with Σ_{l} = l. More precisely

Σ := {(l, v) ∈ ℍP^{n} × ℍ^{n+1} | v ∈ l}, π(l, v) = l.

We shall be concerned with maps f : M → ℍP^{n} from a surface into the projective space. To f we associate the bundle L := f^{∗}Σ, whose fibre over x is f(x) ⊂ ℍ^{n+1} = {x} × ℍ^{n+1}. The bundle L is a line subbundle of the product bundle

H := M × ℍ^{n+1}.

Conversely, every line subbundle L of H over M determines a map f : M → ℍP^{n} by f(x) := L_{x}. We obtain an identification

{Maps \atop f : M → ℍP^{n}}  ↔  Line subbundles L ⊂ H = M × ℍ^{n+1}

All natural constructions for quaternionic vector spaces extend, fibre-wise, to operations in the category of quaternionic vector bundles. For example, a subbundle L of a vector bundle H induces a quotient bundle H/L with fibres H_{x}/L_{x}. However, given two quaternionic vector bundles V_{1}, V_{2}, the homomorphism bundle Hom(V_{1}, V_{2}) is merely a real vector bundle.

Example 4. Over ℍP^{n} we have the product bundle H = ℍP^{n} × ℍ^{n+1} and, inside it, the tautological subbundle Σ. Then

T ℍP^{n} ≅ Hom(Σ, H/Σ),

see (2.1).

Example 5 (and Definition). Let L be a line subbundle of H = M × ℍ^{n+1} corresponding to f : M → ℍP^{n}. Then, by Proposition 1,

f^{∗}T ℍP^{n} ≅ Hom(L, H/L),

and the differential of f = L should be viewed as a 1-form on M with values in Hom(L, H/L):

δ ∈ Ω^{1}(Hom(L, H/L)).  (3.1)

We now endow quaternionic vector bundles with additional structure. A complex quaternionic vector bundle is a pair (V, J) consisting of a quaternionic vector bundle V and a section J ∈ Γ(End(V)) with

J^{2} = −I,

see Section 1.

Example 6. Given f : M → ℍ, ∗df = Ndf, the quaternionic line bundle L = M × ℍ has a complex structure given by

Jv := Nv.

With the use of Lemma 1 we obtain

Proposition 2. Let L ⊂ H = M × ℍ^{2} be an immersed oriented surface in ℍP^{1} with derivative δ ∈ Ω^{1}(Hom(L, H/L)). Then there exist unique complex structures on L and H/L, denoted by J, J̃, such that for all x ∈ M

J̃δ(T_{x}M) = δ(T_{x}M) = δ(T_{x}M)J,
J̃δ = δJ,

and J is compatible with the orientation induced by δ : T_{x}M → δ(T_{x}M).

Definition. A line subbundle L ⊂ H = M × ℍ^{n+1} over a Riemann surface M is called conformal or a holomorphic curve in ℍP^{n}, if there exists a complex structure J on L such that

∗δ = δJ.

From the proposition we see: If L is an immersed holomorphic curve in ℍP^{1}, i.e. if δ is in addition injective, such that δ(T M) ⊂ Hom(L, H/L) is a real subbundle of rank 2, then there is also a complex structure J̃ ∈ Γ(End(H/L)) such that

∗δ = δJ = J̃δ.  (3.2)

A Riemann surface immersed into ℍP^{1} is a holomorphic curve if and only if the complex structures given by the proposition are compatible with the complex structure given on M in the sense of (3.2).

Example 7. Let f : M → ℍ be a conformally immersed Riemann surface with right normal vector R, and let L be the line bundle corresponding to

[\begin{matrix}f \\ 1 \end{matrix}] : M → ℍP^{1}.

Then J (\begin{matrix}f \\ 1 \end{matrix}) = − (\begin{matrix}f \\ 1 \end{matrix}) R defines a complex structure with

δJ = ∗δ.

Hence (L, J) is a holomorphic curve. Conversely, if (L, J) is a holomorphic curve, then

J (\begin{matrix}f \\ 1 \end{matrix}) = − (\begin{matrix}f \\ 1 \end{matrix}) R for some R : M → ℍ, and f is conformal with right normal vector R.

Let (V, J) be a complex quaternionic vector bundle over the Riemann surface M. We decompose

Hom_{ℝ}(T M, V) = KV ⊕ K̄V,

where

KV := {ω : T M → V | ∗ ω = Jω},
K̄V := {ω : T M → V | ∗ ω = −Jω}.

Definition. A holomorphic structure on (V, J) is a quaternionic linear map

D : Γ(V) → Γ(K̄V)

such that for all ψ ∈ Γ(V) and λ : M → ℍ

D(ψλ) = (Dψ)λ + \frac{1}{2}(ψdλ + Jψ ∗ dλ).  (3.3)

A section ψ ∈ Γ(V) is called holomorphic if Dψ = 0, and we put

H^{0}(V) = ker D ⊂ Γ(V).

Remark 1. For a better understanding of this, note that for complex-valued λ the anti-ℂ-linear part (the K̄-part) of dλ is given by ¯∂λ = \frac{1}{2}(dλ + i ∗ dλ). In fact,

(dλ + i ∗ dλ)(JX) = ∗dλ(X) − i dλ(X) = −i(dλ + i ∗ dλ)(X).

A holomorphic structure is a generalized ¯∂-operator. Equation (3.3) is the only natural way to make sense of a product rule of the form D(ψλ) = (Dψ)λ + ψ ¯∂λ.

Example 8. Any given J ∈ End(ℍ^{n}), J^{2} = −1, turns H = M × ℍ^{n} into a complex quaternionic vector bundle. Then Γ(H) = {ψ : M → ℍ^{n}}, and

Dψ := \frac{1}{2}(dψ + J ∗ dψ)

is a holomorphic structure.

Example 9. If L is a complex quaternionic line bundle and ϕ ∈ Γ(L) has no zeros, then there exists exactly one holomorphic structure D on (L, J) such that ϕ becomes holomorphic. In fact, any ψ ∈ Γ(L) can be written as ψ = ϕμ with μ : M → ℍ, and our only chance is

Dψ := \frac{1}{2}(ϕdμ + Jϕ ∗ dμ).  (3.4)

This, indeed, satisfies the definition of a holomorphic structure.

4. The Mean Curvature Sphere

Let M be a Riemann surface. Let H := M × ℍ^{2} denote the product bundle over M, and let S : M → End(ℍ^{2}) ∈ Γ(End(H)) with S^{2} = −I be a complex structure on H. We split the differential according to type:

dψ = d′ψ + d′′ψ,

where d′ and d′′ denote the ℂ-linear and anti-linear components, respectively. In general d(Sψ) ≠ Sdψ, and we decompose further:

d′ = ∂ + A, d′′ = ¯∂ + Q,

where

∂(Sψ) = S∂ψ, ¯∂(Sψ) = S ¯∂ψ,
AS = −SA, QS = −SQ.

Then ¯∂ defines a holomorphic structure and ∂ an anti-holomorphic structure on H, while A and Q are tensorial:

A ∈ Γ(K End_{−}(H)), Q ∈ Γ(K̄ End_{−}(H)).  (4.1)

We find

SdS = 2(Q + A),  (4.2)
Q = \frac{1}{4}(SdS − ∗dS), A = \frac{1}{4}(SdS + ∗dS).  (4.3)

Remark 2. 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 × ℂ) ⊕ (M × ℂ). Therefore A and Q measure the deviation from the “complex case”.

We now consider an immersed holomorphic curve L ⊂ H in ℍP^{1} with derivative δ = δ_{L} ∈ Ω^{1}(Hom(L, H/L)). Then there exist complex structures J on L and J̃ on H/L such that

∗δ = δJ = J̃δ.

We want to extend J and J̃ to a complex structure of H, i.e. to find an

S ∈ Γ(End(H))

such that

SL = L, S|_{L} = J, πS = J̃π.

Note that this implies πdS(ψ) = π(d(Sψ) − Sdψ) = δJψ − J̃δψ = 0, and therefore

dSL ⊂ L.  (4.4)

The existence of S is clear: Write H = L ⊕ L′ for some complementary bundle L′. Identify L′ with H/L using π, and define S|_{L} := J, S|_{L′} := J̃. Since L′ is not unique, S is not unique. It is a very important feature of the 4-dimensional case that we can add a natural condition to single out a unique extension:

Theorem 1. Let L ⊂ H = M × ℍ^{2} be a holomorphic curve immersed into ℍP^{1}. Then there exists a unique complex structure S on H such that

SL = L, dSL ⊂ L,  (4.5)
∗δ = δ ∘ S = S ∘ δ,  (4.6)
Q|_{L} = 0.  (4.7)

S is a family of 2-spheres, a sphere congruence in classical terms. Because S_{p}L_{p} = L_{p} the sphere S_{p} goes through L_{p} ∈ ℍP^{1}, while dSL ⊂ L (or, equivalently, δS = Sδ) implies it is

tangent to L in p, see Example 5. In an affine coordinate system [\begin{matrix}f \\ 1 \end{matrix}] = L the sphere S_{p} has

the same mean curvature vector as f : M → ℝ^{4} = ℍ at p. This motivates the

Definition. S is called the mean curvature sphere (congruence) of L. The differential forms A, Q ∈ Ω^{1}(End(H)) are called the Hopf fields of L.

Remark 3. Equations (4.5), (4.6) imply dψ + S ∗ dψ ∈ Γ(L) for ψ ∈ Γ(L), whence d′′ = ¯∂ + Q = \frac{1}{2}(d + S ∗ d) leaves L invariant. Hence an immersed holomorphic curve in ℍP^{1} is a holomorphic subbundle of (H, S, d′′) and, in particular, is a holomorphic quaternionic vector bundle itself.

We now collect some information about the Hopf fields and the mean curvature sphere congruence S : M → Z.

Lemma 2.

d(A + Q) = 2(Q ∧ Q + A ∧ A).

Lemma 3. Let L ⊂ H be an immersed surface and S a complex structure on H stabilizing L such that dSL ⊂ L. Then Q_{|L} = 0 is equivalent to AH ⊂ L.

Definition. For a quaternionic vector space or bundle V of rank n and an endomorphism A ∈ End(V) we define

< A >:= \frac{1}{4n} trace_{ℝ} A,

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

Proposition 3. The mean curvature sphere S of an immersed Riemann surface 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 [2].

28 D. Ferus: Conformal Geometry of Surfaces in S^{4} and Quaternions

5. Willmore Surfaces Throughout this section M denotes a compact surface. The set

Z = {S ∈ End(ℍ^{2}) | S^{2} = −I}

of oriented 2-spheres in ℍP^{1} is a submanifold of End(ℍ^{2}) with

T_{S}Z = {X ∈ End(ℍ^{2}) | XS = −SX}, ⊥_{S} Z = {Y ∈ End(ℍ^{2}) | Y S = SY }. Here we use the (indefinite) inner product < A, B >:= < AB > = \frac{1}{8} trace_{ℝ}(AB). Definition. The energy functional of a map S : M → Z of a Riemann surface M is defined by

E(S) := ∫_{M} < dS ∧ ∗dS > .

Critical points S of this functional with respect to variations of S are called harmonic maps from M to Z. Proposition 4. S is harmonic if and only if the Z-tangential component of d ∗ dS vanishes:

(d ∗ dS)^{T} = 0.  (5.1)

This condition is equivalent to any of the following:

d(S ∗ dS) = 0,  (5.2)
d ∗ A = 0,  (5.3)
d ∗ Q = 0.  (5.4)

We now consider the case where S is the mean curvature sphere of an immersed holomorphic curve. We decompose dS into the Hopf fields. Lemma 4.

< dS ∧ ∗dS > = 4(< A ∧ ∗A > + < Q ∧ ∗Q >),  (5.5)
< dS ∧ SdS > = 4(< A ∧ ∗A > − < Q ∧ ∗Q >).  (5.6)

Proposition 5. The integral

deg S := \frac{1}{π} ∫_{M} < A ∧ ∗A > − < Q ∧ ∗Q >

is a topological invariant of S.

Therefore,

E(S) = 4 ∫_{M} < A ∧ ∗A > + < Q ∧ ∗Q >= 8 ∫_{M} < A ∧ ∗A > −4π deg S,

and for variational problems the energy functional can be replaced by the integral of < A ∧ ∗A >.

Definition. Let L be a compact immersed holomorphic curve in ℍP^{1} with Hopf field A. The Willmore functional of L is defined as

W (L) := \frac{1}{π} ∫_{M} < A ∧ ∗A > .

If we vary the immersion L : M → ℍP^{1}, it will in general not remain a holomorphic curve. On the other hand, any immersion induces a complex structure J on M such that with respect to this it is a holomorphic curve, see Proposition 2. Critical points of W with respect to such variations are called Willmore surfaces.

Computation of the first variation of W yields

Theorem 2. (Ejiri [3], Rigoli [8]) An immersed holomorphic curve L is Willmore if and only if its mean curvature sphere S is harmonic.

6. Metric and Affine Conformal Geometry

We consider the metric extrinsic geometry of f : M → ℝ^{4} in relation to the quantities associated to

L := [\begin{matrix}f \\ 1 \end{matrix}] : M → ℍP^{1}.

For brevity we write < ., . > instead of < ., . >_{ℝ}. Let N, R denote the left and right normal vector of f : M → ℍ, i.e.

∗df = Ndf = −dfR.

Then

Proposition 6. (i) The second fundamental form II(X, Y) = (X · df(Y))^{⊥} of f is given by

II(X, Y) = \frac{1}{2}(∗df(Y)dR(X) − dN(X) ∗ df(Y)).  (6.1)

(ii) The mean curvature vector H = \frac{1}{2} trace II, the Gaussian curvature K and the normal curvature K^{⊥} are given by

H̄df = \frac{1}{2}(∗dR + RdR), dfH̄ = −\frac{1}{2}(∗dN + NdN)  (6.2)
K|df|^{2} = \frac{1}{2}(< ∗dR, RdR > + < ∗dN, NdN >)  (6.3)
K^{⊥}|df|^{2} = \frac{1}{2}(< ∗dR, RdR > − < ∗dN, NdN >)  (6.4)

(iii) We obtain

(|H|^{2} − K − K^{⊥})|df|^{2} = \frac{1}{4}| ∗ dR − RdR|^{2}

In particular, if f : M → Im ℍ = ℝ^{3} then K^{⊥} = 0, and the classical Willmore integrand is given by

(|H|^{2} − K)|df|^{2} = \frac{1}{4}| ∗ dR − RdR|^{2}.  (6.5)

The Hopf fields are given by

Proposition 7.

4 ∗ Q = G (\begin{matrix}dN + N ∗ dN & 0 \\ −2dH + w & 0 \end{matrix}) G^{−1}, 4 ∗ A = G (\begin{matrix}0 & 0 \\ w & dR + R ∗ dR \end{matrix}) G^{−1},

where G = (\begin{matrix}1 & f \\ 0 & 1 \end{matrix}), and w = dH + H ∗ dfH + R ∗ dH − H ∗ dN.

Proposition 8. The Willmore integrand is given by

< A ∧ ∗A > = \frac{1}{16}|RdR − ∗dR|^{2} = \frac{1}{4}(|H|^{2} − K − K^{⊥})|df|^{2}.

For f : M → ℝ^{3}, this is the classical integrand \frac{1}{4}(|H|^{2} − K)|df|^{2}.

We finally express the Euler-Lagrange equation d ∗ A = 0 for Willmore surfaces in affine coordinates.

Proposition 9.

d ∗ A = \frac{1}{4}G (\begin{matrix}0 & 0 \\ dw & 0 \end{matrix}) G^{−1} = (\begin{matrix}−fdw & −fdwf \\ dw & dwf \end{matrix}) .

with w = dH + R ∗ dH + \frac{1}{2}H(NdN − ∗dN). Therefore, f is Willmore if and only if dw = 0.

Example 10. (Willmore Cylinder) Let γ : ℝ → Im ℍ be a unit-speed curve, and f : ℝ^{2} → ℍ the cylinder defined by

f(s, t) = γ(s) + t

with the conformal structure J \frac{∂}{∂s} = \frac{∂}{∂t}. Then using Proposition 9, we obtain, after some computation, that f is (non-compact) Willmore if and only if

\frac{1}{2}κ^{3} + κ′′ − κτ^{2} = 0, (κ^{2}τ)′ = 0.

This is exactly the condition that γ be a free elastic curve.

7. Twistor Projections

Let E ⊂ H := M × ℍ^{2} = M × ℂ^{4} be a complex (not a quaternionic) line subbundle over a Riemann surface M with complex structure J_{E} induced from right multiplication by i on ℍ^{2}. We define δ_{E} ∈ Ω^{1}(Hom(E, H/E)) by δ_{E}ϕ := π_{E}dϕ, and call E a holomorphic curve in ℂP^{3}, if

∗δ_{E} = δ_{E}J_{E}.

This is equivalent to the fact that the holomorphic structure

d′′ψ = \frac{1}{2}(dψ + i ∗ dψ)  (7.1)

of H maps Γ(E) into itself, and hence induces a holomorphic structure on the complex line bundle E. A complex line bundle E ⊂ H induces a quaternionic line bundle

L = Eℍ = E ⊕ Ej ⊂ H.

The complex structure J_{E} 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) ⊂ H induces a complex line bundle

E := {ϕ ∈ L | Jϕ = ϕi}.

Definition. We call (L, J) the twistor projection of E, and E the twistor lift of (L, J).

Remark 4. As in the quaternionic case, any map f : M → ℂP^{3} induces a complex line bundle E, where the fibre over p is f(p), and vice versa. 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 ℂP^{3} → ℍP^{1}.

Theorem 3. Let E ⊂ H be a a complex line subbundle over a Riemann surface M, and (L, J) its twistor projection. Assume L to be immersed. Then E is a holomorphic curve in ℂP^{3} if and only if (L, J) is a holomorphic curve in ℍP^{1} with A|_{L} = 0.

Given a surface conformally immersed into ℝ^{4}, 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. Using results of Section 6 in connection with Proposition 7, one can show that the condition A|_{L} = 0 is essentially the super-conformality:

Theorem 4. A conformally immersed Riemann surface f : M → ℍ = ℝ^{4} is super-conformal

if and only if [\begin{matrix}f \\ 1 \end{matrix}] : M → ℍP^{1} or [f̄ ; 1 ] : M → ℍP^{1} is the twistor projection of a holomorphic

curve in ℂP^{3}.

8. Bäcklund Transforms of Willmore Surfaces

In this section we shall describe a method to construct new Willmore surfaces from a given one. The construction depends on the choice of a point ∞, 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. Let f : M → ℍ 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 (on the universal covering of M). Assume that g : M → ℍ is an immersion with

dg = \frac{1}{2}w.  (8.1)

We want to show that g is again a Willmore surface called a Bäcklund transform of f. Using this name, we refer to the fact that in a given category of surfaces we construct new examples from old ones by solving an ODE (8.1), similar to the classical Bäcklund transforms of Ksurfaces, see Tenenblat [9]. We denote the associated to g by a subscript (.)_{g}, and want to prove dw_{g} = 0. The computation of w_{g} can be done under the weaker assumption (8.2), which holds in the case above.

Proposition 10. Let f, g : M → ℍ be immersions such that

df ∧ dg = 0.  (8.2)

Then f and g induce the same conformal structure on M, and

N_{g} = −R,  (8.3)
dg(2dH_{g} − w_{g}) = −wdf.  (8.4)

If f is Willmore, and g is defined by (8.1), then

dg(2df + 2dH_{g} − w_{g}) = 2dgdf + dg(2dH_{g} − w_{g}) = (2dg − w)df = 0.

Hence

w_{g} = 2d(f + H_{g}),  (8.5)

and g is Willmore, too. Now assume that h := g − H is again an immersion. Then,

2dh ∧ df = (2dg − 2dH) ∧ df = (w − 2dH) ∧ df = 0.

Proposition 10 applied to (h, f) instead of (f, g) then says

−w_{h}dh = df(2dH − w) = df(2dH − 2dg) = −2dfdh.

We find w_{h} = 2df, whence h is again a Willmore surface. We call g a forward, and h a backward Bäcklund transform of f. h can be obtained without reference to g by integrating d(g − H) = \frac{1}{2}w − dH. Note that f is a forward Bäcklund transform of h because df = \frac{1}{2}w_{h}, and is also a backward transform of g because df = \frac{1}{2}w_{g} − dH_{g}, see (8.5). The concept of Bäcklund transformations depends on the choice of affine coordinates. The following theorem clarifies this situation.

Theorem 5. Let L be a Willmore surface in ℍP^{1}. Choose non-zero β ∈ (ℍ^{2})^{∗}, a ∈ ℍ^{2} such that < β, a > = 0. Then d < β, ∗Aa > = 0 = d < β, ∗Qa >. If g, h : M → ℍ ⊂ ℍP^{1} are immersions that satisfy

dg = 2 < β, ∗Aa >, dh = 2 < β, ∗Qa >,

they are again Willmore surfaces, called forward respectively backward Bäcklund transforms of L. The free choice of β implies that there is a whole S^{4} of such pairs of Bäcklund transforms. (Different choices of a result in Moebius transforms g → gλ, or h → hλ, for a constant λ.)

We can now proceed from g with another forward Bäcklund transform. To do so, we must integrate \frac{1}{2}w_{g} = d(f + H_{g}). But, up to a translational constant, this yields

f̃:= f + H_{g}.  (8.6)

Using Proposition 7, we now get for f̃, and similarly for the twofold backward Bäcklund transform f̂

(f̃ ; 1) ∈ ker A, (f̂ ; 1) ℍ ⊃ image Q.

But this means that away from the zeros of A or Q the 2-step Bäcklund transforms of a Willmore surface L in ℍP^{1} can be described simply as L̃ = ker A or L̂ = image Q. In particular there are no periods arising. It can be shown that (as a consequence of holomorphicity) the kernel of A defines a non-singular line bundle L̃ even at the zeros of A:

Proposition 11. Let L be a Willmore surface in ℍP^{1}, and A ≢ 0 on each component of M. Then there exists a unique line bundle L̃ ⊂ H such that on an open dense subset of M we have:

L̃ = ker A, and H = L ⊕ L̃.

A similar assertion holds for image Q.

Assuming that L̃ is immersed, its invariants can be obtained directly from L. E.g. for the Q-Hopf field we get

Theorem 6. For the 2-step Bäcklund transform L̃ of L we have

Q̃ = A.  (8.7)

Hence L̃ is again a Willmore surface.

We obtain a chain of Bäcklund transforms

⋯ → f̂ → h → f → g → f̃ → ⋯
∥  ∥  ∥
→  L̂  →  L  →  L̃  →

Of course, the chain may break down if we arrive at non-immersed surfaces, or it may close up. Taking the two-step backward transform of L̃, we get image Q̃ = image A = L. Hence ˆL̃ = L. We remark that the above results similarly apply to the backward two-step Bäcklund transformation L → L̂ = image Q. As a corollary of (8.7) and its analog Â = Q we obtain

Theorem 7.

ˆL̃ = L = ˜ L̂.

9. Willmore Surfaces in S^{3}

Let < ., . > be an indefinite hermitian inner product on ℍ^{2}. Then the set of isotropic lines < l, l >= 0 defines an S^{3} ⊂ ℍP^{1}, while the complementary 4-discs are hyperbolic 4-spaces. Let L be a line bundle with mean curvature sphere S. We look at the adjoint map M → Z, p ↦ S_{p}^{∗} with respect to < ., . >. It is easily seen that S = S^{∗} implies that L is isotropic. Conversely, if L is isotropic, then S^{∗} satisfies the characterization of S given in Theorem 1.

Proposition 12. An immersed holomorphic curve L in ℍP^{1} is isotropic, i.e. a surface in S^{3}, if and only if S = S^{∗}.

On the other hand, a 2-sphere S ∈ Z intersects the isotropic S^{3} orthogonally, if and only if it represents totally geodesic 2-planes in the complementary hyperbolic 4-spaces. The orthogonality can be described using affine coordinates with isotropic point ∞ and the reflexion ℍ → ℍ, x ↦ −x̄ at Im ℍ = S^{3}. We get

Proposition 13. A 2-sphere S ∈ Z intersects the hyperbolic 4-spaces determined by an indefinite inner product in hyperbolic 2-planes if and only if S^{∗} = −S.

Now let L be a connected Willmore surface in S^{3} ⊂ ℍP^{1}, where S^{3} is the isotropic set of an indefinite hermitian form on ℍ^{2}. Then its mean curvature sphere satisfies

S^{∗} = S.

Let us assume that A ≢ 0, and let L̃ = ker A and L̂ = image Q be the 2-step Bäcklund transforms of L. Then S^{∗} = S implies L̂ = L̃. Using this one shows that −S satisfies the requirements for S̃, and hence S̃ = −S. We now turn to the 1-step Bäcklund transform of L. If dF = 2 ∗ A, then

d(F + F^{∗}) = 2 ∗ A + 2 ∗ A^{∗} = 2 ∗ A − 2 ∗ Q = −dS.

Because S^{∗} = S, we can choose suitable initial conditions for F such that

F + F^{∗} = −S.  (9.1)

We now use affine coordinates with L = [\begin{matrix}f \\ 1 \end{matrix}]. Then the lower left entry g of F is a Bäcklund

transform of f, and from S^{∗} = S we get g + ḡ = H. Using the properties of Bäcklund transforms we compute the mean curvature sphere S_{g} and find S_{g}^{∗} = −S_{g}. Hence the mean curvature spheres of g intersect S^{3} orthogonally, and therefore are hyperbolic planes. We know 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 “Bäcklund transform” is constant, which may be considered as a degenerate minimal surface. In general g will be singular in the (isolated) zeros of dg = \frac{1}{2}w, but minimal elsewhere. The converse is also true:

Theorem 8. (Richter [7]) Let < ., . > be an indefinite hermitian product on ℍ^{2}. Then the isotropic lines form an S^{3} ⊂ ℍP^{1}, while the two complementary discs inherit complete hyperbolic metrics. Let L be a Willmore surface in S^{3} ⊂ ℍP^{1}. Then a suitable forward Bäcklund transform of L is hyperbolic minimal. Conversely, an immersed holomorphic curve that is hyperbolic minimal is Willmore, and a suitable backward Bäcklund transformation is a Willmore surface in S^{3}.

10. Spherical Willmore Surfaces in ℍP^{1}

In this section we sketch a proof of the following theorem of Montiel, which generalizes an earlier result of Bryant [2] for Willmore spheres in S^{3}.

Theorem 9. (Montiel [4]) A Willmore sphere in ℍP^{1} is a twistor projection of a holomorphic or anti-holomorphic curve in ℂP^{3}, or, in suitable affine coordinates, corresponds to a minimal surface in ℝ^{4}.

The material differs from what we have treated so far: The theorem is global, and therefore requires global methods of proof,imported from complex function theory. Let E be a complex vector bundles. We keep the symbol J ∈ End(H) for the endomorphism given by multiplication with the imaginary unit i. We denote by Ē the bundle where J is replaced by −J. If < ., . > is a hermitian metric on E, then

Ē → E^{∗} = E^{−1}, ψ →< ψ, . >

is an isomorphism of complex vector bundles. Also note that for complex line bundles E_{1}, E_{2} the bundle Hom(E_{1}, E_{2}) is again a complex line bundle. There is a powerful integer invariant for complex line bundles E over a compact Riemann surface, the degree, classifying these bundles up to isomorphism. It can be defined as the integrated curvature form ω of a complex connection, or, equivalently, as the algebraic number of zeros of a generic section ϕ ∈ Γ(E):

deg(E) := \frac{1}{2π} ∫_{M} ω, = ord ϕ := ∑_{ϕ(p)=0} ind_{p} ϕ.

The degree satisfies

deg(Ē) = deg E^{−1} = − deg E,
deg(E_{1} ⊗ E_{2}) = deg E_{1} + deg E_{2}.

Example 11. Let M be a compact Riemann surface of genus g, and E its tangent bundle, viewed as a complex line bundle. Using a complex connection, we compute its degree to find deg(E) = χ(M) = (2 − 2g). For the canonical bundle K := E^{−1} we therefore get

deg(K) = 2g − 2.

Now, let (E, ¯∂) be a holomorphic complex vector bundle. We denote by H^{0}(E|_{U}) the vector space of holomorphic sections over U. If E is a complex line bundle with holomorphic structure, and ψ ∈ H^{0}(E)\{0}, then the zeros of ψ 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 the proof of the Montiel theorem we shall apply the concepts of degree and holomorphicity to several complex bundles obtained from quaternionic ones. We relate these concepts.

Definition. If (L, J) is a complex quaternionic line bundle, then E_{L} := {ψ ∈ L | Jψ = ψi} is a complex line bundle. We define deg L := deg E_{L}.

If L_{1}, L_{2} are complex quaternionic line bundles, and E_{i} := E_{L_{i}}, then Hom_{+}(L_{1}, L_{2}) is isomorphic to Hom_{ℂ}(E_{1}, E_{2}) as a complex vector bundle. In particular

deg Hom_{+}(L_{1}, L_{2}) = − deg L_{1} + deg L_{2}.

We now consider an immersed holomorphic curve L ⊂ H = M × ℍ^{2} in ℍP^{1} 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. The complex structure ¯∂ on T M, and the (quaternionic) holomorphic structures ¯∂ on H and ∂ on H̄ induce a holomorphic structure on

K End_{−}(H) = K Hom_{+}(H̄, H) = K Hom_{ℂ}(H̄, H).

If L is Willmore, then d ∗ A = 0, and this implies ¯∂A = 0, and A is holomorphic:

A ∈ H^{0}(K End_{−}(H)) = H^{0}(K Hom_{+}(H̄, H)).

As a consequence, either A ≡ 0, or the zeros of A are isolated, and there exists a line bundle L̃ ⊂ H such that L̃ = ker A away from the zeros of A. For local ψ ∈ Γ(L̃) and holomorphic Y ∈ H^{0}(T M) we have

\underbrace{¯∂A}_{=0}(Y)ψ = ¯∂(\underbrace{A(Y)ψ}_{=0}) − A(Y)∂ψ.

Therefore L̃ is invariant under ∂, like L is invariant under ¯∂, see Remark 3. As above, we get a holomorphic structure on the complex line bundle K Hom_{+}(H̄/L̃, L) and A defines a holomorphic section of this bundle:

A ∈ H^{0}(K Hom_{+}(H̄/L̃, L)).

We turn to the

Proof of Theorem 9. If A ≡ 0 or Q ≡ 0, then L is a twistor projection by Theorem 4. Otherwise we have the line bundle L̃, and similarly a line bundle L̂ that coincides with the image of Q almost everywhere.

Proposition 14. We have the following holomorphic sections of complex holomorphic line bundles:

A ∈ H^{0}(K Hom_{+}(H̄/L̃, L)), Q ∈ H^{0}(K Hom_{+}(H/L, ¯ L̂)),
δ_{L} ∈ H^{0}(K Hom_{+}(L, H/L)), AQ ∈ H^{0}(K^{2} Hom_{+}(H/L, L)),
and if AQ = 0 then  δL̃ ∈ H^{0}(K Hom_{+}(¯L̃, H̄/L̃)).

We proved the statement about A, the other proofs are similar. The degree formula then yields

ord δ_{L} = deg K − deg L + deg H/L
ord(AQ) = 2 deg K − deg H/L + deg L
= 3 deg K − ord δ_{L}
= 6(g − 1) − ord δ_{L}.

For M = S^{2}, i.e. g = 0, we get ord(AQ) < 0, whence AQ = 0. Then L̃ = L̂, and

ord A = deg K + deg H/L̃ + deg L
ord Q = deg K − deg H/L − deg L̃
ord δL̃ = deg K + deg L̃ − deg H/L̃.

Addition yields

ord δL̃ + ord Q + ord A = 3 deg K − deg H/L + deg L
= 4 deg K − ord δ_{L} = −8 − ord δ_{L}.

It follows that ord δL̃ < 0, i.e. δL̃ = 0, and L̃ is d-stable, hence constant in H = M × ℍ^{2}. From AS = −SA = 0 we conclude SL̃ = L̃. Therefore all mean curvature spheres of L pass through the fixed point L̃. Choosing affine coordinates with L̃ = ∞, all mean curvature spheres are affine planes, and L corresponds to a minimal surface in ℝ^{4}.

References

[1] Burstall, Francis; Ferus, Dirk; Leschke, Katrin; Pedit, Franz; Pinkall, Ulrich: Conformal Geometry of Surfaces in S^{4} and Quaternions. To appear.

[2] Bryant, Robert: A duality theorem for Willmore surfaces. J. Differential Geom. 20 (1984), 23–53.

[3] Ejiri, Norio: Willmore Surfaces with a Duality in S^{N}(1). Proc. Lond. Math. Soc., III Ser. 57, No.2 (1988), 383–416.

[4] Montiel, Sebastián: Spherial Willmore Surfaces in the Four-Sphere. Preprint 1998.

[5] 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)

[6] Pedit, Franz; Pinkall, Ulrich; et al.: Quaternionic Holomorphic Geometry and Willmore surfaces in S^{4}. To appear.

[7] Richter, Jörg: Conformal Maps of a Riemannian Surface into the Space of Quaternions. Dissertation, Berlin 1997.

[8] Rigoli, Marco: The conformal Gauss map of Submanifolds of the Moebius Space. Ann. Global Anal. Geom. 5, No.2 (1987), 97–116.

[9] Tenenblatt, Keti: Transformations of Manifolds and Applications to Differential Equations. Chapman & Hall/CRC Press 1998.