Existence of stable H-surfaces in cones and their representation as radial graphs

TL;DR

This paper proves the existence of stable H-surfaces within cones that can be represented as radial graphs under certain conditions, extending the understanding of surfaces with prescribed mean curvature in conic regions.

Contribution

It establishes the existence of least energy H-surfaces in cones and characterizes when these surfaces can be expressed as radial graphs based on geometric and monotonicity conditions.

Findings

Existence of least energy H-surfaces spanning Jordan curves in cones.

Conditions under which H-surfaces can be represented as radial graphs.

Extension of the Plateau problem to conic regions with prescribed mean curvature.

Abstract

In this paper we study the Plateau problem for disk-type surfaces contained in conic regions of $\mathbb{R}^{3}$ and with prescribed mean curvature $H$. Assuming a suitable growth condition on $H$, we prove existence of a least energy $H$-surface $X$ spanning an arbitrary Jordan curve $\Gamma$ taken in the cone. Then we address the problem of describing such surface $X$ as radial graph when the Jordan curve $\Gamma$ admits a radial representation. Assuming a suitable monotonicity condition on the mapping $\lambda\mapsto\lambda H(\lambda p)$ and some strong convexity-type condition on the radial projection of the Jordan curve $\Gamma$, we show that the $H$-surface $X$ can be represented as a radial graph.