We consider operators of the form $\Delta+aK$ on a Riemannian surface. Such operators naturally appear when considering the stability operator of a minimal surface in a 3-manifold. In particular, when studying the stability of minimal surfaces, a natural problem is to derive geometric properties of the surface from the positivity of the operator. In this talk we will prove that the positivity of $\Delta+aK$ on a Riemannian surface (with additional hypotheses when $a\le\frac{1}{4}$) imply that the surface is conformally equivalent to $\mathbb{C}$ or $ \mathbb{C}^*$, and in the second case we will prove that the metric is flat. We shall see that our statements are sharp, improving former results on the subject. This is a joint work with Pierre Bérard.

We shall see how to prove isoperimetric inequalities on submanifolds of the Euclidean space, using mass transportation methods. We obtain a sharp weighted isoperimetric inequality and a nonsharp classical inequality. The proof relies on the description of a solution of the problem of Monge when the initial measure is supported in a submanifold and the final one supported in a linear subspace of the same dimension.