Stable constant mean curvature surfaces with free boundary in slabs

Abstract

We study stable constant mean curvature (CMC) hypersurfaces $\Sigma$ in slabs in a product space $M\times\r,$ where $M$ is an orientable Riemannian manifold. We obtain a characterization of stable cylinders and prove that if $\Sigma$ is not a cylinder then it is locally a vertical graph. Moreover, in case $M$ is $\h^n,\r^n$ or $\s_+^n$ and each of its boundary components is embedded then $\Sigma$ is rotationally invariant. When $M$ has dimension 2 and Gaussian curvature bounded from below by a positive constant $\kappa,$ we prove there is no stable CMC with free boundary connecting the boundary components of a slab of width $l>4\pi/\sqrt{3\kappa}.$ We also show that a stable capillary surface of genus 0 in a warped product $[0,l]\times_f M$ where $M=\r^2, \h^2$ or $\s^2,$ is rotationally invariant. Finally, we prove that a stable closed CMC surface in $M\times\s^1(r),$ where $M$ is a surface with Gaussian curvature bounded from below by a positive constant $\kappa$ and $\s^1(r)$ the circle of radius $r,$ lifts to $M\times\r$ provided $r>4/\sqrt{3\kappa}.$