Entire spacelike radial graphs with prescribed mean curvature in the Lorentz--Minkowski space

TL;DR

This paper investigates the existence, uniqueness, and geometric inequalities of entire spacelike hypersurfaces with prescribed mean curvature in Lorentz--Minkowski space, focusing on star-shaped graphs asymptotic to light cones.

Contribution

It establishes new existence and uniqueness results for spacelike hypersurfaces with prescribed mean curvature and introduces a Willmore-type inequality in this setting.

Findings

Proves existence and uniqueness of entire spacelike hypersurfaces with prescribed mean curvature.

Establishes a Willmore-type inequality for such hypersurfaces.

Shows non-existence of certain radial graphs with mean curvature in L^p.

Abstract

In this paper we address the existence and uniqueness of entire spacelike hypersurfaces in the Lorentz--Minkowski space $\mathbb{L}^{m+1}$ with prescribed mean curvature that are star-shaped with respect to a point and asymptotic to a light cone. We also establish a Willmore-type inequality and prove a non-existence result for spacelike radial graphs asymptotic to the light cone whose mean curvature belongs to $L^p$ for $1 \leq p\leq m$, in particular in the case of compactly supported mean curvature.