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 free boundary stable CMC surfaces.