Rigidity of Free Boundary Surfaces in Compact 3-Manifolds with Strictly Convex Boundary

TL;DR

This paper establishes a sharp geometric bound for free boundary minimal surfaces in convex 3-manifolds, characterizing Euclidean balls when the bound is achieved, thus extending classical rigidity results.

Contribution

It provides a new analogue of Toponogov's theorem in three dimensions, with sharp bounds and rigidity characterizations for free boundary minimal surfaces.

Findings

Sharp upper bound for boundary length in terms of genus and components

Characterization of Euclidean 3-balls when bounds are saturated

Bounds on surface area in convex bodies, with equality only for Euclidean balls

Abstract

In this paper we obtain an analogue of Toponogov theorem in dimension 3 for compact manifolds $M^3$ with nonnegative Ricci curvature and strictly convex boundary $\partial M$. Here we obtain a sharp upper bound for the length $L(\partial\Sigma)$ of the boundary $\partial\Sigma$ of a free boundary minimal surface $\Sigma^2$ in $M^3$ in terms of the genus of $\Sigma$ and the number of connected components of $\partial\Sigma$, assuming $\Sigma$ has index one. After, under a natural hypothesis on the geometry of $M$ along $\partial M$, we prove that if $L(\partial\Sigma)$ saturates the respective upper bound, then $M^3$ is isometric to the Euclidean 3-ball and $\Sigma^2$ is isometric to the Euclidean disk. In particular, we get a sharp upper bound for the area of $\Sigma$, when $M^3$ is a strictly convex body in $\mathbb R^3$, which is saturated only on the Euclidean 3-balls (by the Euclidean disks). We also consider similar results for stationary stable surfaces.