2018/02/19 by Souam, Rabah
#49Q10 #53A10 #53C42 #76B45 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1802.06848
We study stable constant mean curvature (CMC) hypersurfaces Σ in slabs in a product space M×\r, where M is an orientable Riemannian manifold. We obtain a characterization of stable cylinders and prove that if Σ is not a cylinder then it is locally a vertical graph. Moreover, in case M is \hn,\rn or \s+n and each of its boundary components is embedded then Σ is rotationally invariant. When M has dimension 2 and Gaussian curvature bounded from below by a positive constant κ, we prove there is no stable CMC with free boundary connecting the boundary components of a slab of width l>4π/√(3κ). We also show that a stable capillary surface of genus 0 in a warped product [0,l]×f M where M=\r2, \h2 or \s2, is rotationally invariant. Finally, we prove that a stable closed CMC surface in M×\s1(r), where M is a surface with Gaussian curvature bounded from below by a positive constant κ and \s1(r) the circle of radius r, lifts to M×\r provided r>4/√(3κ).