Prescribed curvature measure problem in hyperbolic space

TL;DR

This paper addresses the prescribed curvature measure problem in hyperbolic space, establishing existence results for star-shaped k-convex bodies with specific curvature measures through advanced PDE regularity estimates.

Contribution

It provides the first existence results for prescribed curvature measures in hyperbolic space using new C^2 regularity estimates for nonlinear PDEs.

Findings

Existence of star-shaped k-convex bodies with prescribed curvature measures.

Development of crucial C^2 regularity estimates for fully nonlinear PDEs in hyperbolic space.

Extension of prescribed curvature measure theory to hyperbolic geometry.

Abstract

The problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this paper, we consider prescribed curvature measure problem in hyperbolic space. We obtain the existence of star-shaped k-convex bodies with prescribed (n-k)-th curvature measures (k<n) by establishing crucial C^2 regularity estimates for solutions to the corresponding fully nonlinear PDE in the hyperbolic space.