Convex Spacelike Hypersurfaces of Constant Curvature in de Sitter Space

TL;DR

This paper proves the existence of complete, convex spacelike hypersurfaces with prescribed constant curvature in de Sitter space, extending the asymptotic Plateau problem and establishing a duality with hyperbolic space.

Contribution

It introduces a general framework for solving the curvature prescription problem in de Sitter space, including a duality with hyperbolic space and existence results for various curvature functions.

Findings

Existence of smooth solutions for a broad class of curvature functions.

Uniqueness results for certain cases when l=1 or l=2.

Duality between de Sitter and hyperbolic space problems.

Abstract

We show that for a very general and natural class of curvature functions (for example the curvature quotients $(\sigma_n/\sigma_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurface in de Sitter space satisfying $f(\kappa) = \sigma \in (1,\infty)$ with a prescribed compact future asymptotic boundary $\Gamma$ at infinity has at least one smooth solution (if l = 1 or l = 2 there is uniqueness). This is the exact analogue of the asymptotic plateau problem in Hyperbolic space and is in fact a precise dual problem. By using this duality we obtain for free the existence of strictly convex solutions to the asymptotic Plateau problem for $\sigma_l = \sigma$; $1\leq l < n$ in both deSitter and Hyperbolic space.