A holomorphic representation formula

for parabolic hyperspheres

Vicente Cortés ^{∗} Mathematisches Institut Universität Bonn Beringstraße 1 D-53115 Bonn

July 15, 2001

Abstract A holomorphic representation formula for special parabolic hyperspheres is given. Keywords: affine hyperspheres, special Kähler manifolds MSC 2000: 53A15, 53C26

Introduction

It was noticed by Blaschke that parabolic spheres in affine 3-space admit parametrisations in terms of holomorphic functions (of one variable). This is related to the fact that the Monge-Ampère equation governing parabolic hyperspheres is completely integrable in dimension two, a fact already known to Monge. The purpose of this note is to derive an explicit formula describing special parabolic hyperspheres in affine (2n + 1)-space in terms of a holomorphic function of n variables.

∗e-mail: vicente@math.uni-bonn.de This work was supported by SFB 256 “Nichtlineare partielle Differentialgleichungen” (Universität Bonn).

1 Special parabolic affine hyperspheres

Let me briefly recall the notion of special parabolic affine hypersphere [BC]. We consider ℝ^{m+1} as affine space with standard connection denoted by ˜∇ and parallel volume form vol. A hypersurface is given by an immersion φ : M → ℝ^{m+1} of an m-dimensional connected manifold. We assume that M admits a transversal vector field ξ and that m > 1. This induces on M the volume form ν = vol(ξ, …), a torsionfree connection ∇, a quadratic covariant tensor field g, an endomorphism field S (shape tensor) and a one-form θ such that

˜∇_{X}Y = ∇_{X}Y + g(X, Y)ξ ,
˜∇_{X}ξ = SX + θ(X)ξ .  (1.1)

Let us call the data (∇, g, S, θ) the Gauß-Weingarten data induced by the transversal vector field ξ. We will assume that g is nondegenerate and, hence, is a pseudo-Riemannian metric on M. This condition does not depend on the choice of ξ. According to Blaschke [B], once the orientation of M is fixed, there is a unique choice of transversal vector field ξ such that ν coincides with the metric volume form vol^{g} and ∇ν = 0. This particular choice of transversal vector field is called the affine normal and the corresponding geometric data (g, ∇) are called Blaschke metric and induced connection. Notice that, for the affine normal, θ = 0 and S is computable from (g, ∇) (Gauß equations). Henceforth we use always the affine normal as transversal vector field.

Definition 1 The hypersurface φ : M → ℝ^{m+1} is called a parabolic (or improper) hypersphere if the affine normal is parallel, ˜∇ξ = 0. It is called special if there exists an almost complex structure J on M which is skew symmetric with respect to the Blaschke metric g and such that the 2-form ω := g(J·, ·) is parallel with respect to the induced connection ∇. Such an almost complex structure J is called compatible.

Notice that ˜∇ξ = 0 ⇔ S = 0 ⇔ ∇ is flat. It was proven in [BC] that any parabolic twodimensional sphere with (positive or negative) definite Blaschke metric is special and that a compatible almost complex structure on a special parabolic hypersphere is necessarily integrable. In fact, we proved the following stronger result, Theorem 1 below.

Recall that a special Kähler manifold (M, J, g, ∇) is a (pseudo-)Kähler manifold (M, J, g) endowed with a flat torsionfree connection ∇ such that ∇J is symmetric and ∇ω = 0, where ω = g(J·, ·) is the Kähler form.

Theorem 1 [BC] Let φ : M → ℝ^{m+1} be a special parabolic hypersphere with Blaschke metric g, induced connection ∇, compatible almost complex structure J and canonical twoform ω = g(J·, ·). Then (M, J, g, ∇) is a special Kähler manifold. Conversely, any simply connected special Kähler manifold (M, J, g, ∇) admits an immersion φ : M → ℝ^{m+1}, which is a special parabolic hypersphere with Blaschke metric g, induced connection ∇ and compatible almost complex structure J. The immersion φ is unique up to a unimodular affine transformation of ℝ^{m+1}.

2 The holomorphic representation formula

It was proven in [ACD] that any special Kähler manifold is locally defined by a holomorphic function, as follows. Let F be a holomorphic function on a domain (i.e. a connected open set) U ⊂ ℂ^{n} such that the (real) matrix

Im ∂^{2}F is invertible,  (2.1)

where ∂^{2}F denotes the holomorphic Hessian of F. Let us denote by M_{F} ⊂ T^{∗}ℂ^{n} the image of the holomorphic section dF : U → T^{∗}U ⊂ T^{∗}ℂ^{n}. It is a complex Lagrangian submanifold with respect to the standard complex symplectic structure Ω = ∑ dz^{i} ∧dw_{i}, where (z^{1}, …, z^{n}, w_{1}, …, w_{n}) are canonical coordinates of T^{∗}ℂ^{n}. We denote its complex structure by J. Using the nondegeneracy condition (2.1), it is shown in [ACD] that the Hermitian form γ := \sqrt{−1}Ω(·,¯·) is nondegenerate on M_{F} and, hence, induces a (pseudo-) Kähler metric g = Re γ|_{M_{F}}. It is also shown that a torsionfree connection ∇ on M_{F} can be defined by the condition that the real parts x^{i} := Re z^{i} and y_{j} := Re w_{j} are ∇-affine functions on M_{F}. In fact, it is shown that (x^{1}, …, x^{n}, y_{1}, …, y_{n}) is a (real) local coordinate system near any point of M_{F} and that the Kähler form ω = g(·, J·) is expressed by the formula ω = 2 ∑ dx^{i} ∧ dy_{i} on M_{F}.

Theorem 2 [ACD] Let F be a holomorphic function satisfying the nondegeneracy condition (2.1) on a domain U ⊂ ℂ^{n}. Then (M_{F} , J, g, ∇), defined above, is a special Kähler manifold and any special Kähler manifold is locally of this form.

It is noticed in [BC] that combining Theorem 1 and Theorem 2 we can associate a parabolic hypersphere to any holomophic function F defined on a simply connected domain U ⊂ ℂ^{n} and satisfying the nondegeneracy condition (2.1). However, the proof of Theorem 1 makes use of the Fundamental Theorem of affine differential geometry [DNV] (the generalisation of Radon’s theorem [R] to higher dimensions) and does not involve any explicit parametrisation of the immersion φ : M → ℝ^{2n+1} realising a simply connected special Kähler manifold (M, J, g, ∇) of real dimension 2n as parabolic hypersphere. The aim is now to provide an explicit formula, in terms of the holomorphic function F, for the realisation of (M_{F} , J, g, ∇) as a parabolic hypersphere φ_{F} : M_{F} ≅ U → ℝ^{2n+1}.

We will not restrict ourselves to functions F defined on simply connected domains U ⊂ ℂ^{n}. More generally, we consider a ‘multivalued’ function defined on an arbitrary domain U ⊂ ℂ^{n}. Or, in other words, a (univalued) function defined on some Riemann domain Ũ over U. A Riemann domain over U is a holomorphic (unramified) covering π : Ũ → U. Any holomorphic function F on Ũ defines a holomorphic Lagrangian immersion

ϕ : Ũ → T^{∗}U ⊂ T^{∗}ℂ^{n} , ϕ(p) := dF ∘ (π_{∗}|T_{p}Ũ)^{−1} , p ∈ Ũ .  (2.2)

Let us denote by J the complex structure of Ũ. Pulling back the canonical coordinates of T^{∗}ℂ^{n} to Ũ we obtain holomorphic functions

z̃^{i} := ϕ^{∗}z^{i} and w̃_{j} := ϕ^{∗}w_{j}

on Ũ. The holomorphic functions z̃^{i} form a local holomorphic coordinate system near any point of Ũ. We use the compact notation

z̃:= (z̃^{1}, …, z̃^{n}) , Fz̃ = (F_{}_{z̃^{1}}, …, F_{}_{z̃^{n}}) = (\frac{∂F}{∂z̃^{1}} , …, \frac{∂F}{∂z̃^{n}}) , Fz̃z̃ = ∑ F_{}_{z̃^{k}}z̃^{k} etc.

Let ∂^{2}F be the Hessian of F with respect to (the flat torsionfree holomorphic connection defined by) the coordinate system z̃. We call F nondegenerate if Im ∂^{2}F is invertible. Then, as before, g := Re ϕ^{∗}γ is a pseudo-Kähler metric and we can define a flat torsionfree connection ∇ by the condition that the functions x̃^{i} := Re z̃^{i} and ỹ_{j} := Re w̃_{j} are ∇-affine functions on Ũ. We also put ũ^{i} := Im z̃^{i} and ṽ_{j} := Im w̃_{j}. Let us abbreviate M(F) := (Ũ, J, g, ∇) and define an immersion φ_{F} : Ũ → ℝ^{2n+1} by the formula

φ_{F} := (Re z̃, Re Fz̃, 2Im F − 2(Re Fz̃)Im z̃)  (2.3)
= (x̃^{1}, …, x̃^{n}, ỹ_{1}, …, ỹ_{n}, 2Im F − 2 ∑(ỹ_{k})ũ^{k}) .

Theorem 3 Let F be a nondegenerate holomorphic function defined on a Riemann domain Ũ. Then M(F) = (Ũ, J, g, ∇), defined above, is a special Kähler manifold with Kähler form ω = g(·, J·) = 2 ∑ dx̃^{i} ∧ dỹ_{i}. The immersion φ_{F} : Ũ → ℝ^{2n+1} defined by (2.3) is, with respect to the volume form vol := 2^{n} det on ℝ^{2n+1}, a special parabolic hypersphere with affine normal ξ = ∂_{2n+1}, Blaschke metric g, induced connection ∇ and compatible almost complex structure J. It is unique up to unimodular affine transformations of ℝ^{2n+1}.

Proof: The first statement is a slight generalisation of the first part of Theorem 2, with essentially the same proof. The uniqueness of φ_{F} follows, as in the proof of Theorem 1, from the uniqueness statement of the Fundamental Theorem of affine differential geometry. It suffices to prove that φ_{F} is a parabolic hypersphere with Blaschke metric g, induced connection ∇ and compatible almost complex structure J. Let us compute the Gauß-Weingarten data (∇^{v}, g^{v}, S^{v}, θ^{v}), see (1.1), induced by the transversal vector field v = ∂_{2n+1} (the ‘vertical’ vector field). It is immediate that S^{v} = 0 and θ^{v} = 0. We compute ∇^{v} and g^{v} for the coordinate vector fields

∂_{}_{x̃^{i}} = ∂_{i} + \frac{∂f}{∂x̃^{i}} ∂_{2n+1} , ∂ỹ_{j} = ∂_{n+j} + \frac{∂f}{∂ỹ_{j}} ∂_{2n+1} ,

where

f := 2Im F − 2(Re Fz̃)Im z̃ = 2Im F − 2 ∑ ỹ_{k}ũ^{k}

is the last component of φ_{F}. The covariant derivatives with respect to the connection ˜∇ of ℝ^{2n+1} are given by

˜∇_{∂_{x̃^{i}}}∂_{}_{x̃^{j}} = \frac{∂^{2}f}{∂_{}_{x̃^{i}}∂_{}_{x̃^{j}}} v , ˜∇_{∂_{x̃^{i}}}∂ỹ_{j} = ˜∇_{∂ỹ_{j}} ∂_{}_{x̃^{i}} = \frac{∂^{2}f}{∂_{}_{x̃^{i}}∂ỹ_{j}} v , ˜∇_{∂ỹ_{i}}∂ỹ_{j} = \frac{∂^{2}f}{∂ỹ_{i}∂ỹ_{j}} v .

This shows that the coordinate vector fields ∂_{}_{x̃^{i}} and ∂ỹ_{j} are parallel for the connection ∇^{v}, so it coincides with ∇. Now θ = 0 implies ∇ν^{v} = ∇^{v}ν^{v} = 0 for the volume form ν^{v} = vol(v, …). Moreover, we see that g^{v} = Hess^{∇}(f) = ∇^{2}f. Claim 1 g^{v} = g. The claim, to be proven below, implies that v is the affine normal and, hence, that g^{v} = g is the Blaschke metric. Let us see why. The Riemannian volume of the (pseudo-)Kähler manifold M(F) with Kähler form ω = g(J·, ·) = 2 ∑_{i=1}^{n} dx̃^{i} ∧ dỹ_{i} is given by

vol^{g} = (−1)^{n(n−1)/2} \frac{ω^{n}}{n!} = 2^{n}dx̃^{1} ∧ … ∧ dx̃^{n} ∧ dỹ_{1} ∧ … ∧ dỹ_{n} = 2^{n} det(v, …) = ν^{v} ,

if we choose the orientation defined by ν^{v}. (Notice that dx̃^{1} ∧ … ∧ dx̃^{n} ∧ dỹ_{1} ∧ … ∧ dỹ_{n} = (−1)^{n(n−1)/2}dx̃^{1} ∧dỹ_{1} ∧…∧dx̃^{n} ∧dỹ_{n}.) This shows that vol^{g} coincides with the ∇-parallel volume form ν^{v}. So v is the affine normal and, hence, φ_{F} is a parabolic hypersphere with Blaschke metric g and induced connection ∇. The fact that M(F) is a special Kähler manifold entails that J is skew symmetric with respect to g and ∇ω = 0. Therefore, the complex structure J is compatible, in the sense of Definition 1, with the data (g, ∇). It remains to prove Claim 1. For the calculations we will use the next lemma.

Lemma 1 The partial derivatives of the functions ũ^{i} and ṽ_{j} on M(F) with respect to the ∇-affine coordinates (x̃^{1}, …, x̃^{n}, ỹ_{1}, …, ỹ_{n}) satisfy the following equations:

∑_{k}(ũ_{x̃^{i}}^{k}(ṽ_{k})ỹ_{j} − ũk ; ỹ_{j}(ṽ_{k})_{}_{x̃^{i}}) = δ_{i}^{j} ,

∑_{k} ũ_{x̃^{i}}^{k}(ṽ_{k})_{}_{x̃^{j}} = ∑_{k} ũ_{x̃^{j}}^{k}(ṽ_{k})_{}_{x̃^{i}} ,

∑_{k} ũk ; ỹ_{i}(ṽ_{k})ỹ_{j} = ∑_{k} ũk ; ỹ_{j}(ṽ_{k})ỹ_{i}

ũ_{x̃^{j}}^{i} = −(ṽ_{j})ỹ_{i} ,

ũi ; ỹ_{j} = ũj ; ỹ_{i} ,

(ṽ_{i})_{}_{x̃^{j}} = (ṽ_{j})_{}_{x̃^{i}} .

Proof: Pulling back the symplectic form Ω of T^{∗}ℂ^{n} by means of the Lagrangian immersion ϕ : M(F) → T^{∗}ℂ^{n} defined in (2.2), we obtain the equation ϕ^{∗}Ω = 0. Decomposing it into real and imaginary parts yields the lemma. Let us return to the proof of Theorem 3. First we observe that

∂_{}_{x̃^{i}}Im F = Im ∂_{}_{x̃^{i}}F = ∑_{j} Im (\frac{∂z̃^{j}}{∂x̃^{i}} ∂_{}_{z̃^{j}}F) = ∑_{j} Im ((δ_{i}^{j} + \sqrt{−1}ũ_{x̃^{i}}^{j})F_{}_{z̃^{j}}) = ṽ_{i} + ∑_{j} ũ_{x̃^{i}}^{j}ỹ_{j}

and

∂ỹ_{j}Im F = Im ∂ỹ_{j}F = ∑_{k} Im (\frac{∂z̃^{k}}{∂ỹ_{j}} ∂_{}_{z̃^{k}}F) = ∑_{k} Im (\sqrt{−1}ũk ; ỹ_{j}Fz̃_{k}) = ∑_{k} ũk ; ỹ_{j} ỹ_{k} .

The second derivatives of Im F are now easily computed with the help of Lemma 1:

∂_{x̃^{i}x̃^{j}}^{2}Im F = (ṽ_{i})_{}_{x̃^{j}}+∑_{k} ũ_{x̃^{i}x̃^{j}}^{k} ỹ_{k} , ∂_{x̃^{i}ỹ_{j}}^{2}Im F = ∑_{k} ũ_{ỹ_{j}x̃^{i}}^{k}ỹ_{k} , ∂_{ỹ_{i}ỹ_{j}}^{2}Im F = ũi ; ỹ_{j}+∑_{k} ũ_{ỹ_{i}ỹ_{j}}^{k} ỹ_{k} .

Using this and Lemma 1 one can now evaluate g^{v} = ∇^{2}f:

g^{v}(∂_{}_{x̃^{i}}, ∂_{}_{x̃^{j}}) = ∂_{x̃^{i}x̃^{j}}^{2}f = 2((ṽ_{i})_{}_{x̃^{j}} + ∑_{k} ũ_{x̃^{i}x̃^{j}}^{k} ỹ_{k}) − 2 ∑_{k} ũ_{x̃^{i}x̃^{j}}^{k} ỹ_{k} = 2(ṽ_{i})_{}_{x̃^{j}} ,
g^{v}(∂_{}_{x̃^{i}}, ∂ỹ_{j}) = ∂_{x̃^{i}ỹ_{j}}^{2}f = 2 ∑_{k} ũ_{ỹ_{j}x̃^{i}}^{k}ỹ_{k} − 2(ũ_{x̃^{i}}^{j} + ∑_{k} ũ_{ỹ_{j}x̃^{i}}^{k}ỹ_{k}) = −2ũ_{x̃^{i}}^{j} ,  (2.4)
g^{v}(∂ỹ_{i}, ∂ỹ_{j}) = ∂_{ỹ_{i}ỹ_{j}}^{2}f = 2(ũi ; ỹ_{j} + ∑_{k} ũ_{ỹ_{i}ỹ_{j}}^{k} ỹ_{k}) − 2(ũj ; ỹ_{i} + ũi ; ỹ_{j} + ∑_{k} ũ_{ỹ_{i}ỹ_{j}}^{k} ỹ_{k}) = −2ũi ; ỹ_{j} .

Notice that in virtue of (2.4) we have:

(ũ^{i})_{}_{x̃^{j}} = (ũ^{j})_{}_{x̃^{i}} .  (2.5)

Let us compare this with g. The simplest way to compute g is using the fact that g = ω∘J, where ω = 2 ∑ dx^{i} ∧ dy_{i} is the Kähler form and we consider g and ω as isomorphisms T M → T^{∗}M (insertion of a vector in the first argument). It is easier to work with the inverse metric g^{−1} = J^{−1} ∘ ω^{−1} = −J ∘ ω^{−1} = ω^{−1} ∘ J^{∗}. Notice that

ω^{−1} = \frac{1}{2} ∑ ∂ỹ_{i} ∧ ∂_{}_{x̃^{i}} , J^{∗}dx̃^{i} = −dũ^{i} and J^{∗}dỹ_{j} = −dṽ_{j} .

Let us evaluate g^{−1} with the help of these formulas and Lemma 1:

g^{−1}(dx̃^{i}, dx̃^{j}) = −ω^{−1}(dũ^{i}, dx̃^{j}) = −\frac{1}{2}ũi ; ỹ_{j} ,
g^{−1}(dx̃^{i}, dỹ_{j}) = −ω^{−1}(dũ^{i}, dỹ_{j}) = \frac{1}{2}ũ_{x̃^{j}}^{i} ,
g^{−1}(dỹ_{i}, dỹ_{j}) = −ω^{−1}(dṽ_{i}, dỹ_{j}) = \frac{1}{2}(ṽ_{i})_{}_{x̃^{j}} .

Comparing with the formulas for g^{v} and using Lemma 1 and (2.5) this proves that g^{−1}g^{v} = id and, hence, that g = g^{v}. This completes the proof of Claim 1 and Theorem 3.

References

[ACD] D. V. Alekseevsky, V. Cortés and C. Devchand, Special complex manifolds, available as preprint 2000-14, Max-Planck-Institut für Mathematik, math.DG/9910091.

[BC] O. Baues and V. Cortés, Realisation of special Kähler manifolds as parabolic spheres, to appear in Proc. Amer. Math. Soc., available as preprint 613, SFB 256, Bonn, math.DG/9911079.

[B] W. Blaschke, Vorlesungen über Differentialgeometrie II. Affine Differentialgeometrie, Grundlehren der Mathematischen Wissenschaften VII, Springer Verlag, Berlin 1923.

[DNV] F. Dillen, K. Nomizu and L. Vrancken, Conjugate connections and Radon’s theorem in affine differential geometry, Monatsh. Math. 109 (1990), 221-235.

[R] J. Radon, Die Grundgleichungen der affinen Flächentheorie, Leipziger Berichte 70 (1918), 91-107.