Strictly Locally Convex Radial Graphs of Prescribed Curvature and Boundary in Space Forms

TL;DR

This paper develops a method to find strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms, providing $C^2$ estimates and existence results using degree theory.

Contribution

It introduces new $C^2$ a priori estimates and existence proofs for convex hypersurfaces with prescribed curvature in space forms, under subsolution assumptions.

Findings

Established $C^2$ a priori estimates for solutions.

Proved existence of hypersurfaces with prescribed curvature.

Applied degree theory to geometric PDEs.

Abstract

We obtain $C^2$ a priori estimates for solutions of the nonlinear second-order elliptic equation related to the geometric problem of finding a strictly locally convex hypersurface with prescribed curvature and boundary in a space form. Under the assumption of a strictly locally convex subsolution, we establish existence results by using degree theory arguments.