Complete constant mean curvature surfaces and bernstein type theorems in $m^2 x r$

JM Espinar Universite Paris VII H Rosenberg Instituto de Matematica Pura y Aplicada

Differential Geometry mathscidoc:1609.10127

Journal of Differential Geometry, 82, (3), 611-628, 2009
In this paper we study constant mean curvature surfaces  in a product space, M2 × R, where M2 is a complete Riemannian manifold. We assume the angle function ν = hN, @ @t i does not change sign on . We classify these surfaces according to the infimum c() of the Gaussian curvature of the projection of . When H 6= 0 and c() ≥ 0, then  is a cylinder over a complete curve with curvature 2H. If H = 0 and c() ≥ 0, then  must be a vertical plane or  is a slice M2 ×{t}, or M2 ≡ R2 with the flat metric and  is a tilted plane (after possibly passing to a covering space). When c() < 0 and H > p −c()/2, then  is a vertical cylinder over a complete curve of M2 of constant geodesic curvature 2H. This result is optimal. We also prove a non-existence result concerning complete multigraphs in M2 × R, when c(M2) < 0.
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@inproceedings{jm2009complete,
  title={COMPLETE CONSTANT MEAN CURVATURE SURFACES AND BERNSTEIN TYPE THEOREMS IN $M^2 x R$},
  author={JM Espinar, and H Rosenberg},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911185718999978784},
  booktitle={Journal of Differential Geometry},
  volume={82},
  number={3},
  pages={611-628},
  year={2009},
}
JM Espinar, and H Rosenberg. COMPLETE CONSTANT MEAN CURVATURE SURFACES AND BERNSTEIN TYPE THEOREMS IN $M^2 x R$. 2009. Vol. 82. In Journal of Differential Geometry. pp.611-628. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911185718999978784.
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