Hypersurfaces of constant scalar curvature in hyperbolic space with prescribed asymptotic boundary at infinity

TL;DR

This paper proves the existence of smooth hypersurfaces with constant scalar curvature in hyperbolic space that match a prescribed boundary at infinity, extending previous results to all curvature values using new techniques.

Contribution

It introduces novel methods to solve a fully nonlinear elliptic PDE for all curvature values, generalizing prior restricted results in hyperbolic space.

Findings

Existence of hypersurfaces with prescribed asymptotic boundary and constant scalar curvature.

Solution of a degenerate Dirichlet problem for all curvature values.

Development of new second order a priori estimates for the PDE.

Abstract

This article concerns a natural generalization of the classical asymptotic Plateau problem in hyperbolic space. We prove the existence of a smooth complete hypersurface of constant scalar curvature with a prescribed asymptotic boundary at infinity. The desired hypersurface is constructed as the limit of constant scalar curvature graphs (with respect to vertical geodesics) over a fixed compact domain in a horosphere, and the problem is thus reduced to solving a Dirichlet problem for a fully nonlinear elliptic partial differential equation which is degenerate along the boundary. Previously, the result was known only for a restricted range of curvature values. Now, in this article, by introducing some new techniques, we are able to solve the Dirichlet problem for all possible curvature values. The main ingredient is the establishment of the crucial second order a priori estimates for admissible solutions.