Discutiremos la geometría de los dominios $\Omega$ en el plano tales que, bajo diferentes hipótesis, admiten una solución del problema clásico sobreterminado $\Delta u + f(u) = 0$, $u>0$ en $\Omega$, $u=0$, $\partial u/\partial ν=c$ en $\partial\Omega$. Este es un trabajo conjunto con P. Sicbaldi.

Although the recent complete solution of the Willmore conjecture depends on the min-max theory of minimal surfaces, some partial results can be obtained by using the isoperimetric problem on 3-manifolds of positive Ricci curvature and, in particular, on the real projective space. Using this tool, we will present a proof of the Willmore conjecture for antipodal invariant tori in the 3-sphere and for the case of tori with a central symmetry in Euclidean 3-space.

Dado un dominio plano y una funcion $f$, consideramos la ecuación $\Delta u + f(u) =0$, $u>0$ en $\Omega$ con condiciones de Dirichlet y Neumann en la frontera, simultáneamente, $u=0$, $u_\nu=c$ en $\partial\Omega$. Si el dominio es acotado, Serrin demuestra que $\Omega$ es un disco y $u$ es radial. Para dominios no acotados hay muchas cuestiones abiertas y presentamos algunos resultados obtenidos con Pieralberto Sicbaldi y relacionados con la teoría de superficies de curvatura media constante: dominios con topología finita, teorema del semiplano, etc.

We review some stability and area minimizing properties for minimal and constant mean curvature surfaces in the euclidean 3-space. We will present the case of complete surfaces, free boundary and the prescribed symmetries one. In particular, we will prove that area minimizing surfaces in some quotients of $\mathbb{R}^3$ are planar.

A surface $S$ in a complete $3$-manifold $M$ is area-minimizing mod $2$ if it has least area among all surfaces, orientable or nonorientable in the same homology class. These surfaces present a rich and interesting geometry, even in flat or positively curved $3$-manifolds. For instance, if $M$ is flat, Fischer-Colbrie and Schoen, Do Carmo and Peng, and Pogorelov proved that complete two-sided stable minimal surfaces are flat, but Ross proved that some nonorientable quotient of the classic Schwarz P and D surfaces are estable, and we proved that an area minimizing surface in $\mathbb{R}^2\times S^1$ is either planar or a quotient of the Helicoid. We will review some results about this problem and we will prove that area minimizing surfaces in flat $3$-tori are planar.