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 $\partial M$. 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.