Isoperimetric and stable sets for log-concave perturbations of Gaussian measures

Let $\Omega$ be an open half-space or slab in $\mathbb{R}^{n+1}$ endowed with a perturbation of the Gaussian measure of the form $f(p):=\exp(\omega(p)-c|p|^2)$, where $c>0$ and $\omega$ is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to $\partial\Omega$. In this work we follow a variational approach to show that half-spaces perpendicular to $\partial\Omega$ uniquely minimize the weighted perimeter in $\Omega$ among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spaces parallel or perpendicular to $\partial\Omega$ as the unique stable sets with small singular set and null weighted capacity. Our methods also apply for $\Omega=\mathbb{R}^{n+1}$, which produces in particular the classification of stable sets in Gauss space and a new proof of the Gaussian isoperimetric inequality. Finally, we use optimal transport to study the weighted minimizers when the perturbation term $\omega$ is concave and possibly non-smooth.Comment: final version, to appear in Analysis and Geometry in Metric Space

The classification of complete stable area-stationary surfaces in the Heisenberg group H1\mathbb{H}^1

We prove that any $C^2$ complete, orientable, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group $\mathbb{H}^1$ is either a Euclidean plane or congruent to the hyperbolic paraboloid $t=xy$.Comment: 32 pages, no figures, added reference missed in version

Parabolicity criteria and characterization results for submanifoldsof bounded mean curvature in model manifolds with weights

Let P be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight e h. The aim of this paper is twofold. First, by assuming certain control on the h-mean curvature of P, we establish comparisons for the h-capacity of extrinsic balls in P, from which we deduce criteria ensuring the h-parabolicity or h-hyperbolicity of P. Second, we employ functions with geometric meaning to describe submanifolds of bounded h-mean curvature which are confined into some regions of the ambient manifold. As a consequence, we derive half-space and Bernstein-type theorems generalizing previous ones. Our results apply for some relevant h-minimal submanifolds appearing in the singularity theory of the mean curvature flow