2023/07/07 by Laurent Mazet, Mazet, Laurent, Abraão Mendes +1
Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2307.03624
openalex publication_date 2023/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 II∂ B≥ g|∂ B, there exists a free boundary minimal disk Δ of least area among all free boundary minimal disks in (B,g). Moreover, the area of any such Δ equals to the width of (B,g), Δ has index one, and the length of ∂Δ is bounded from above by 2π. Furthermore, the length of ∂Δ equals to 2π 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.