We prove that the stability index of a compact constant mean curvature (CMC) surface in the Euclidean space or in the unit sphere is bounded from below by a linear function of its genus. We also will discuss some results in the case of free-boundary CMC surfaces in a mean convex body. These results are part of joint works with Darlan de Oliveira.

In the first part of this talk we will show some lower estimates for the first eigenvalue of a minimal hypersurface immersed in H^mxR. As an application we prove that if a complete minimal surface in H^2xR has finite total extrinsic curvature, then such surface has finite Jacobi index. In the second part we will apply the method of Colding-Minicozzi to provide some upper estimates for the first eigenvalue of a complete Riemannian surfaces. We then apply this estimates for minimal surface in H3 and H^2xR. These results are part of a joint work with P. Bérard and Ph. Castillon.