Optimal length estimates for stable CMC surfaces in 3-space forms

TL;DR

This paper establishes an optimal upper bound on the distance from any point to the boundary in stable constant mean curvature surfaces in 3-space forms, extending known results and providing new geometric estimates.

Contribution

It proves an optimal boundary distance estimate for stable CMC surfaces in space forms, generalizing previous results in Euclidean space.

Findings

Distance from a point to boundary is less than π/(2H) in stable CMC surfaces.

The bound is proven to be optimal.

Extension of the estimate to space forms.

Abstract

In this paper, we study stable constant mean curvature $H$ surfaces in $\R^3$. We prove that, in such a surface, the distance from a point to the boundary is less that $\pi/(2H)$. This upper-bound is optimal and is extended to stable constant mean curvature surfaces in space forms.