Compact surfaces with boundary with prescribed mean curvature depending on the Gauss map

TL;DR

This paper studies surfaces with boundary in three-dimensional space whose mean curvature depends on the Gauss map, establishing existence, non-existence, and geometric estimates under various conditions.

Contribution

It introduces a framework for analyzing $ ext{H}$-surfaces with prescribed mean curvature depending on the Gauss map, including existence, non-existence, and geometric bounds results.

Findings

Non-existence of closed $ ext{H}$-surfaces under mild conditions.

Conditions for rotational symmetry when boundary is a circle.

Area estimates based on the height of the surface.

Abstract

Given a $C^1$ function $\mathcal{H}$ defined in the unit sphere $\mathbb{S}^2$, an $\mathcal{H}$-surface $M$ is a surface in the Euclidean space $\mathbb{R}^3$ whose mean curvature $H_M$ satisfies $H_M(p)=\mathcal{H}(N_p)$, $p\in M$, where $N$ is the Gauss map of $M$. Given a closed simple curve $\Gamma\subset\mathbb{R}^3$ and a function $\mathcal{H}$, in this paper we investigate the geometry of compact $\mathcal{H}$-surfaces spanning $\Gamma$ in terms of $\Gamma$. Under mild assumptions on $\mathcal{H}$, we prove non-existence of closed $\mathcal{H}$-surfaces, in contrast with the classical case of constant mean curvature. We give conditions on $\mathcal{H}$ that ensure that if $\Gamma$ is a circle, then $M$ is a rotational surface. We also establish the existence of estimates of the area of $\mathcal{H}$-surfaces in terms of the height of the surface.