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

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 $\mathrm{\partial }M$. Here we obtain a sharp upper bound for the length $L(\mathrm{\partial }\mathrm{\Sigma })$ of the boundary $\mathrm{\partial }\mathrm{\Sigma }$ of a free boundary minimal surface ${\mathrm{\Sigma }}^2$ in $M^3$ in terms of the genus of $\mathrm{\Sigma }$ and the number of connected components of $\mathrm{\partial }\mathrm{\Sigma }$, assuming $\mathrm{\Sigma }$ has index one. After, under a natural hypothesis on the geometry of $M$ along $\mathrm{\partial }M$, we prove that if $L(\mathrm{\partial }\mathrm{\Sigma })$ saturates the respective upper bound, then $M^3$ is isometric to the Euclidean 3-ball and ${\mathrm{\Sigma }}^2$ is isometric to the Euclidean disk. In particular, we get a sharp upper bound for the area of $\mathrm{\Sigma }$, 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.