In this talk, I will study closed embedded minimal hypersurfaces in Riemannian manifold that minimize area among such hypersurfaces : they exist and have index at most 1. I will apply this to minimal surfaces in hyperbolic 3-manifolds.

In this talk, we prove that, in a stable cmc surface in $\mathbb{R}^3$, the intrinsic distance from a point to the boundary is less than $\pi/(2H)$. This estimate is sharp and can be extended to $\mathbb{H}^3$ and $\mathbb{S}^3$.