Stable constant mean curvature surfaces with free boundary in slabs

We study stable constant mean curvature (CMC) hypersurfaces $\mathrm{\Sigma }$ in slabs in a product space $M\times \mathring{,}$ where $M$ is an orientable Riemannian manifold. We obtain a characterization of stable cylinders and prove that if $\mathrm{\Sigma }$ is not a cylinder then it is locally a vertical graph. Moreover, in case $M$ is \h^n,\r^n or ${\text{s}}_{+}^n$ and each of its boundary components is embedded then $\mathrm{\Sigma }$ 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 $l>4\pi \mathrm{/}\sqrt{3\kappa }\mathrm{.}$ 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 ${\text{s}}^2,$ is rotationally invariant. Finally, we prove that a stable closed CMC surface in $M\times {\text{s}}^1(r),$ where $M$ is a surface with Gaussian curvature bounded from below by a positive constant $\kappa$ and ${\text{s}}^1(r)$ the circle of radius $r,$ lifts to M\times\r provided $r>4\mathrm{/}\sqrt{3\kappa }\mathrm{.}$