Strictly Locally Convex Hypersurfaces with Prescribed Curvature and Boundary in Space Forms

TL;DR

This paper establishes existence results for strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms, using a two-step continuity method and degree theory to obtain a priori estimates and solve the curvature equation.

Contribution

It introduces a novel approach combining a two-step continuity process and degree theory to prove existence of convex hypersurfaces with prescribed curvature in space forms.

Findings

Existence of convex hypersurfaces with prescribed Gauss curvature in space forms.

Development of a new a priori estimate technique for curvature equations.

Application of degree theory to curvature problems with boundary conditions.

Abstract

This paper is devoted to $C^2$ a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature equation under the assumption of a strictly locally convex subsolution.