Sharp Area Bounds for Free Boundary Minimal Surfaces in Conformally Euclidean Balls

TL;DR

This paper establishes sharp area bounds for free boundary minimal surfaces within conformally Euclidean balls, extending previous Euclidean results to more general conformal settings and characterizing equality cases.

Contribution

It generalizes known Euclidean area bounds for free boundary minimal surfaces to conformally Euclidean spaces, providing new geometric inequalities and rigidity results.

Findings

Area of free boundary minimal surfaces bounded by conformally Euclidean balls is bounded below by that of a geodesic disk.

Equality occurs only when the surface is a geodesic disk itself.

Results extend previous Euclidean bounds to broader conformal ambient spaces.

Abstract

We prove that the area of a free boundary minimal surface $\Sigma^2 \subset B^n$, where $B^n$ is a geodesic ball contained in a round hemisphere $\mathbb{S}^n_+$, is at least as big as that of a geodesic disk with the same radius as $B^n$; equality is attained only if $\Sigma$ coincides with such a disk. More generally, we prove analogous results for a class of conformally euclidean ambient spaces. This follows work of Brendle and Fraser-Schoen in the euclidean setting.