[Paper Review] Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space

Bo Guan, Huaiyu Jian+1 more|ArXiv.org|Nov 29, 2006

Geometric Analysis and Curvature Flows7 references3 citations

TL;DR

This paper establishes the existence and regularity of entire spacelike hypersurfaces in Minkowski space with prescribed Gauss curvature, using a variational approach to construct admissible maximal solutions satisfying weak spacelike conditions. It proves that for Lipschitz boundary data in dimension two, there exists a unique smooth, strictly convex solution to the Monge-Ampère equation, yielding a complete, smooth spacelike hypersurface with the given curvature function.

ABSTRACT

We are concerned with spacelike convex hypersurfaces of positive constant (K-hypersurfaces) or prescribed Gauss curvature in Minkowski space. Our main purpose is to study entire solutions as well as the Dirichlet problem in bounded domains of the related Monge-Ampere equation.

Motivation & Objective

• To classify entire spacelike K-hypersurfaces with rotational symmetry in Minkowski space, extending prior results on non-uniqueness of isometric embeddings of hyperbolic space.
• To study the asymptotic behavior of entire spacelike hypersurfaces with unbounded principal curvatures by characterizing their tangent cones at infinity.
• To solve the Minkowski-type problem for Gauss curvature on a subset of the unit sphere in Minkowski space, particularly for $ \Omega = \mathbb{H}^n_+ $, by constructing solutions with prescribed boundary data.
• To extend Li's results on the Minkowski problem to allow Lipschitz boundary data in dimension two, which is a more natural geometric assumption than smoothness.
• To establish existence and regularity of weak solutions to the Monge-Ampère equation under the weak spacelike condition $ |Du| \leq 1 $, using a variational framework with admissible maximal solutions.

Proposed method

• Construct rotationally symmetric spacelike hypersurfaces in $ \mathbb{R}^{n,1} $ using ordinary differential equations, analyzing their completeness and asymptotic behavior at infinity.
• Introduce a variational approach to solve the Monge-Ampère equation $ \det D^2 u = \psi(x,u)(1 - |Du|^2)^{\frac{n+2}{2}} $ under the weak spacelike condition $ |Du| \leq 1 $, defining admissible maximal solutions.
• Use the Legendre transform to reframe the Minkowski-type problem in terms of curvature functions defined on the unit ball $ B_1(0) \subset \mathbb{R}^n $, linking the Gauss map to the solution's normal vector.
• Apply Caffarelli’s theory on viscosity solutions and the Evans-Krylov regularity theory to prove that convex viscosity solutions are smooth and strictly convex in the interior.
• Establish boundary estimates using supporting planes and barrier functions, proving that solutions satisfy $ \lim_{t \to 0^+} \frac{v(\hat{y} + t\mathbf{e}) - v(\hat{y})}{t} = -\infty $ at boundary points, which rules out linear growth and ensures strict convexity.
• Use approximation arguments and maximum principle estimates to prove continuity up to the boundary and regularity in the interior, particularly for Lipschitz boundary data in dimension two.

Experimental results

Research questions

• RQ1 Can entire spacelike hypersurfaces of prescribed Gauss curvature be constructed in Minkowski space without assuming boundedness of principal curvatures?
• RQ2 What is the asymptotic behavior of such hypersurfaces at infinity, and how can it be characterized via their tangent cones?
• RQ3 Is it possible to solve the Minkowski-type problem for Gauss curvature on a proper subset of the unit sphere, such as $ \mathbb{H}^n_+ $, with minimal regularity assumptions on the boundary data?
• RQ4 Can the existence of smooth, complete spacelike hypersurfaces be established for Lipschitz boundary data in the two-dimensional case, extending prior results that required smoothness?
• RQ5 How do admissible maximal solutions satisfying $ |Du| \leq 1 $ relate to classical solutions of the Monge-Ampère equation in the spacelike regime?

Key findings

• Rotationally symmetric entire spacelike K-hypersurfaces in $ \mathbb{R}^{n,1} $ are complete with respect to the induced Riemannian metric.
• For $ n=2 $, there exists a unique smooth, strictly convex solution $ v \in C^\infty(\Omega) \cap C^0(\overline{\Omega}) $ to the Monge-Ampère equation with Lipschitz boundary data, satisfying $ \lim_{t \to 0^+} \frac{v(\hat{y} + t\mathbf{e}) - v(\hat{y})}{t} = -\infty $ at boundary points.
• The resulting hypersurface $ M = \text{graph}(v^*) $ is smooth, complete, and strictly convex, with Gauss curvature $ K_M(\mathbf{n}^{-1}(y)) = \eta(y) $ for all $ y \in B_1(0) $.
• The solution $ v $ satisfies the asymptotic condition $ x \cdot Dv(x) - v(x) \to -\infty $ as $ |x| \to \infty $, indicating unbounded principal curvatures.
• The boundary condition $ \varphi \in C^{0,1}(\partial\Omega) $ is necessary and sufficient for the existence of a solution when $ n=2 $, and the solution fails to be $ C^{0,1} $ at points where $ \varphi $ is strictly convex.
• The Minkowski-type problem on $ \Omega = \mathbb{H}^n_+ $ is solvable under the given assumptions, and the solution class includes hypersurfaces with unbounded principal curvatures.