SURFACES OF CONSTANT CURVATURE IN AFFINE SPACE

XXI Congresso UMI Topologia e Geometria Differenziale

Pavia, 3/9/2019

Joint work with Xin Nie

THE HYPERBOLOID

Let us start with a simple object: (a connected component of) the two-sheeted hyperboloid in three-space.

ℍ2 = {(x, y, z) ∈ ℝ3 : x2 + y2 − z2 = − 1, z > 0}

= {v ∈ ℝ3 : ⟨v, v⟩ = − 1, ⟨v, ∂z⟩ < 0},

where we denote the standard bilinear form of signature (2,1) (the Minkowski metric) by ⟨v1, v2⟩ = x1x2 + y1y2 − z1z2

THE HYPERBOLOID

The hyperboloid has some simple properties:

• It is an entire graph (in fact, it is asymptotic to the cone x2+y2=z2);

• Its shape operator is B=id, where B=dN is an endomorphism of the tangent space, and N is the unit normal vector with respect to the Minkowski metric.

Moreover, it is unique (up to Minkowski isometries) under these conditions.

We will look at more general classes of surfaces: 1. Surfaces of constant Gaussian curvature for the Minkowski metric 2. Affine spheres in affine space 3. Surfaces of constant Gaussian affine curvature

1. CONSTANT GAUSSIAN CURVATUREWe say an embedded surface in three-space has constant Gaussian curvature (with respect to the Minkowski metric) if the induced metric is Riemannian and det(B)=K, for some constant K>0.

The problem of classification of entire graphs of constant Gaussian curvature has been studied: Li ’95, Guan-Jian-Schoen ’06, Barbot-Béguin-Zeghib ’11, Bonsante-S.-Smillie ’19.

Analytically, it is equivalent to the Monge-Ampère problem on the disc (here K=1):

det D2u = 1(1 − |z |2 )2

in 𝔻

u |∂𝔻 = φ∥∇u∥ → + ∞ towards ∂𝔻

where the necessary and sufficient condition for existence and uniqueness is that φ is lower semicontinuous and finite on 3 points.

THE DIRICHLET CONDITIONThe Dirichlet-type condition has a geometric interpretation. It says that the surface is asymptotic to the deformation of the cone:

u |∂𝔻 = φ

Dφ = ⋂p∈∂𝔻

Hp,φ(p)

where we are taking the intersection of (at least three) half-spaces

Hp,φ(p) = {z ≥ p ⋅ (x, y) − φ(p)}

2. AFFINE SPHERESThere is another generalisation of the hyperboloid in the context of affine differential geometry.

Given a convex embedded surface Σ in three-space, there is a notion of affine normal field N which is well-defined for the action of the group of volume-preserving affine transformations.

In fact, for a transversal vector field N, we can write the equations:

SA(ℝ3)=SL3(ℝ)⋉ℝ3

DXY = ∇XY + h(X, Y )NDXN = B(X) + τ(X)N

Then N is the affine normal field (which is unique up to sign) if the induced volume form on Σ coincides with the volume form of h, and moreover

Finally, Σ is a proper affine sphere if B is a nonzero multiple of the identity. Equivalently, the affine normals meet at the same point.

τ ≡ 0.ν = ιN det

CHENG-YAU THEOREMHyperbolic affine spheres are those for which the normals meet at a point in the concave side of Σ. This corresponds to B=λid, λ>0.

det D2v = v−4 in Ωv |∂Ω = 0∥∇u∥ → + ∞ towards ∂Ω

Analytically, it reduces to the Monge-Ampère problem on

Theorem (Cheng-Yau ’77): For every proper convex cone C, there exists a unique complete hyperbolic affine sphere with B=id, asymptotic to the boundary of C.

Ω=C∩{z=1}:

In fact, on the disc is a solution for both problems, with zero boundary value, corresponding to the hyperboloid.

u = v = − 1 − |z |2

Or equivalently on the dual convex domain

Ω*=C*∩{z= − 1}:

Examples are the hyperboloid and the Țițeica affine sphere xyz=1, which is asymptotic to the boundary of the first octant.

3. CONSTANT AFFINE CURVATURE

We shall now study a larger class of surfaces, defined by the condition det(B)=K for some positive constant K, which includes both surfaces of constant Gaussian curvature and affine spheres.

Important fact: A convex surface Σ satisfies det(B)=K>0 if and only if the affine normals of Σ, translated at the origin, form a hyperbolic affine sphere (with shape operator λid, λ=λ(K)).

Given a proper convex cone, we say that Σ is an affine (C,λ)-surface if its affine normals have image in the unique complete affine sphere asymptotic to the boundary of C, with shape operator λid.

• Complete affine spheres asymptotic to C are affine (C,λ)-surfaces. • Surfaces of constant Gaussian curvature (for the Minkowski

metric) are precisely the affine (C0,λ)-surfaces, where C0 is the quadratic cone x2+y2<z2. In this case, the affine normal and the Minkowski normal coincide.

MAIN THEOREM

Let us assume: • C is a proper convex cone, and denote Ω=C∩{z=1}, Ω*=C*∩{z=1}; • Ω* satisfies the exterior circle condition, i.e. for every p in the

boundary of Ω*, there is a circle through p which contains Ω*; • is a lower semicontinuous function, finite on at

least three points; • λ is a positive constant.

Then there exists a unique complete affine (C,λ)-surface asymptotic to the boundary of

φ : ∂Ω* → ℝ ∪ {+∞}

Dφ = ⋂p∈∂Ω*

Hp,φ(p)

where the intersection is among the half-spaces

Hp,φ(p) = {z ≥ p ⋅ (x, y) − φ(p)}

ANALYTIC VIEWPOINT

In terms of Monge-Ampère equations, the problem is (here λ=1):

det D2u = v−4 in Ω*u |∂Ω* = φ∥∇u∥ → + ∞ towards ∂Ω*

where v is the Cheng-Yau solution of

{det D2v = v−4 in Ω*v |∂Ω* = 0

Observe that we recover all the solutions of the previous two Monge-Ampère problems: constant Gaussian curvature surfaces (when Ω=Ω* is the disc) and affine spheres (when φ=0).

A SUBTLE POINT

Hence to determine a surface of affine constant curvature, one must specify the domain D to which it is asymptotic and the cone C to which the corresponding affine sphere is asymptotic.

For instance, a domain D which is the intersection of three half-spaces can be obtained as a deformation of different cones, and thus it contains several surfaces asymptotic to its boundary.

On the left, D is seen as a deformation of the quadratic cone, hence it contains a surface of constant Gaussian curvature. On the right, D is seen as a cone itself, hence it contains the Țițeica affine sphere.

COUNTEREXAMPLES

In fact, by a variation of this example, one can produce a proper convex cone C and a deformation D which contains no complete affine (C,λ)-surface.

The exterior circle condition fails on the left picture, and is satisfied on the right instead. Dually, the picture is:

THE EXTERIOR CIRCLE CONDITION

Hence the second cone C, for which there does exists a complete affine (C,λ)-surface asymptotic to the “triangular” deformation D, looks like:

On the other hand, our theorem is not sharp: by a result of Labourie, there exist complete affine (C,λ)-surface asymptotic to deformations of cones C for which the interior circle condition fails.

THANKS FOR YOUR ATTENTION!