Hypersurfaces of constant scalar curvature in hyperbolic space with prescribed asymptotic boundary at infinity (2408.07656v3)

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 hypersurface is constructed as the graph of some smooth function 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, we are able to solve the Dirichlet problem for all possible curvature values by establishing the crucial second order a priori estimates for admissible solutions. This resolves a longstanding problem, and for the proof, since the currently available techniques in the literature do not apply, we have introduced a new reduction technique which we hope could be inspiring for future study in the field.