Complete constant mean curvature surfaces and Bernstein type theorems in $M^2 x R$
By J. Espinar, H. Rosenberg. 2008
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In this paper we study constant mean curvature surfaces Σ in a product space, M 2 × R , where M 2 is a complete Riemannian manifold. We assume the angle function ν = h N, ∂ ∂t i does not change sign on Σ. We classify these surfaces according to the infimum c (Σ) of the Gaussian curvature of the projection of Σ. When H 6 = 0 and c (Σ) ≥ 0, then Σ is a cylinder over a complete curve with curvature 2 H . If H = 0 and c (Σ) ≥ 0, then Σ must be a vertical plane or Σ is a slice M 2 × { t } , or M 2 ≡ R 2 with the flat metric and Σ is a tilted plane (after possibly passing to a covering space). When c (Σ) < 0 and H > p − c (Σ) / 2, then Σ is a vertical cylinder over a complete curve of M 2 of constant geodesic curvature 2 H . This result is optimal. We also prove a non-existence result concerning complete multi-graphs in M 2 × R , when c ( M 2 ) < 0.
Subfields: Applied Mathematics, Astronomy and Astrophysics
View Paper: https://www.semanticscholar.org/paper/e5ba7bf357fa449cda695225ace76222a72bc714
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In Applied Mathematics
In Astronomy and Astrophysics
Citations in the corpus
- Halfspace type theorems for self‐shrinkers (2014, methodology): “2 We split the proof in two cases: the case of the ball, which is simple, and the case of a nondegenerated cylinder, where we use a stability argument following some ideas in [7].”
- The half-space property and entire positive minimal graphs in M x R (2012, background): “In the same spirit, an interesting question is to study those complete embedded minimal hypersurfaces in M × R, whose angle function 〈N, ∂ ∂t 〉 does not change sign; see [5].”