[Paper Review] Regular domains and surfaces of constant Gaussian curvature in three-dimensional affine space

Xin Nie, Andrea Seppi|HAL (Le Centre pour la Communication Scientifique Directe)|Mar 19, 2019

Geometric Analysis and Curvature FlowsMathematics14 references3 citations

TL;DR

This paper establishes the existence and uniqueness of complete surfaces with constant affine Gaussian curvature (CAGC) in 3D affine space, foliating proper regular domains defined by convex cones. By solving a two-step Monge-Ampère equation with lower semicontinuous boundary data and gradient blowup, it generalizes both affine spheres and constant curvature surfaces in Minkowski space, proving that every such domain admits a unique CAGC foliation under exterior circle condition assumptions.

ABSTRACT

Generalizing the notion of domains of dependence in the Minkowski space, we define and study regular domains in the affine space with respect to a proper convex cone. In dimension three, we show that every proper regular domain is uniquely foliated by a particular kind of surfaces with constant affine Gaussian curvature. The result is based on the analysis of a Monge-Ampère equation with extended-real-valued lower semicontinuous boundary condition.

Motivation & Objective

• To unify two classical results: affine spheres in convex cones and constant curvature surfaces in Minkowski space.
• To generalize the Dirichlet problem for Monge-Ampère equations to include extended-real-valued, lower semicontinuous boundary data.
• To establish the existence and uniqueness of complete surfaces with constant affine Gaussian curvature in 3D affine space.
• To characterize the geometric structure of regular domains via CAGC foliations.
• To extend the theory of affine differential geometry to include non-smooth boundary data and singular curvature conditions.

Proposed method

• Formulates a two-step Monge-Ampère equation involving the solution to the standard Monge-Ampère equation (1.1) and a curvature-dependent right-hand side.
• Imposes a gradient blowup condition at the boundary of the domain to ensure completeness of the resulting surface.
• Applies the exterior circle condition on the boundary of the domain to control the growth of the right-hand side and ensure solution regularity.
• Uses a duality construction between the convex cone and its dual to relate the boundary function φ to the geometry of the domain.
• Employs a variational approach and regularity theory for degenerate Monge-Ampère equations with singular right-hand sides.
• Applies results from convex analysis and projective geometry to characterize the domain as a regular domain generated by a CAGC surface.

Experimental results

Research questions

• RQ1 Under what conditions does a proper regular domain in 3D affine space admit a unique foliation by surfaces of constant affine Gaussian curvature?
• RQ2 Can the solution to the Monge-Ampère equation with singular boundary data and gradient blowup be uniquely extended to a complete surface with prescribed CAGC?
• RQ3 How does the geometry of the boundary of the domain (e.g., exterior circle condition) affect the existence and regularity of CAGC surfaces?
• RQ4 What is the relationship between the projectivized dual cone and the existence of complete affine (C,k)-hypersurfaces in the domain?
• RQ5 When does the absence of C1 smoothness in the boundary of the dual cone prevent the existence of a complete CAGC surface?

Key findings

• For any bounded convex domain Ω ⊂ ℝ² satisfying the exterior circle condition and any lower semicontinuous boundary function φ with at least three finite values, there exists a unique lower semicontinuous convex function u solving the two-step Monge-Ampère equation (1.3).
• The domain of definition of the solution u coincides with the convex hull of the set where φ is finite, ensuring geometric consistency with the boundary data.
• The solution u has infinite inner derivatives at every boundary point of its domain, which guarantees the completeness of the associated CAGC surface.
• In the case where the dual cone’s projectivized boundary is a triangle and the boundary curve meets each edge in a non-tangential, non-degenerate way, the domain is uniquely foliated by complete affine (C,k)-surfaces for every k > 0.
• If the boundary of the dual cone meets an edge of the triangle at a single non-smooth point with both tangents pointing inward, no such complete CAGC surface exists.
• The solution u belongs to the class S₀(Ω) if and only if the exterior circle condition holds at each vertex of the domain, ensuring finite inner derivatives at vertices and regularity of the solution.