Radial graphs of constant curvature and prescribed boundary

TL;DR

This paper develops a method to find hypersurfaces with constant curvature and prescribed boundary in Euclidean space by leveraging the theory of fully nonlinear elliptic equations, focusing on radial graphs.

Contribution

It introduces a new approach using subsolutions to construct radial graphs with specified constant curvature and boundary conditions.

Findings

Existence of radial graphs with constant scalar curvature for certain domains.

Application to mean convex domains in spheres with prescribed boundary.

Use of fully nonlinear elliptic equations to solve boundary value problems.

Abstract

In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, using the theory of fully nonlinear elliptic equations. We prove that if the given data admits a suitable radial graph as a subsolution, then we can find a radial graph with constant curvature and that realizes the prescribed boundary. As an application we prove that if $\Omega\subset\mathbb{S}^n$ is a mean convex domain whose closure is contained in an open hemisphere of $\mathbb{S}^n$ then, for $0<R<n(n-1),$ there exists a radial graph of constant scalar curvature $R$ and boundary $\partial\Omega.$