In this paper we prove a rigidity statement for free boundary minimal surfaces produced via min–max methods. More precisely, for any Riemannian metric g on the 3-ball B with non-negative Ricci curvature and \(\textrm{II}_{\partial B}\ge g_{|\partial B}\) , there exists a free boundary minimal disk \(\Delta \) of least area among all free boundary minimal disks in (B, g). We prove that 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.
