Rigidity of min-max minimal disks in $3$-balls with non-negative Ricci curvature

TL;DR

This paper establishes a rigidity property for free boundary minimal disks in 3-balls with non-negative Ricci curvature, linking their area, boundary length, and the geometry of the ambient space.

Contribution

It proves a new rigidity theorem for minimal disks in 3-balls with non-negative Ricci curvature, extending previous results to free boundary minimal surfaces.

Findings

Existence of a least-area free boundary minimal disk in the given setting.

The area of such disks equals the width of the manifold.

The boundary length is bounded above by 2π, with equality characterizing the Euclidean ball.

Abstract

In this paper we prove a rigidity statement for free boundary minimal surfaces produced via min-max methods. More precisely, we prove that for any Riemannian metric $g$ on the 3-ball $B$ with non-negative Ricci curvature and $\mathrm{II}_{\partial B}\ge g_{|\partial B}$, there exists a free boundary minimal disk $\Delta$ of least area among all free boundary minimal disks in $(B,g)$. Moreover, the area of any such $\Delta$ equals to the width of $(B,g)$, $\Delta$ has index one, and the length of $\partial\Delta$ is bounded from above by $2\pi$. Furthermore, the length of $\partial\Delta$ equals to $2\pi$ if and only if $(B,g)$ is isometric to the Euclidean unit ball. This is related to a rigidity result obtained by F.C. Marques and A. Neves in the closed case. The proof uses a rigidity statement concerning half-balls with non-negative Ricci curvature which is true in any dimension.