Stable Constant Mean Curvature Surfaces with Free Boundary in Slabs
Résumé
We study stable constant mean curvature (CMC) hypersurfaces Σ with free boundary 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 Sn+, if each component of ∂Σ 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 boundaryco nnecting 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 CMC immersion of a closed surface in M × S1(r), where M is a surface with Gaussian curvature bounded from below by a positive constant κ and S1(r) thecircle of radius r, lifts to M × R provided r ≥ 4/√3κ.
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