Properly embedded surfaces with prescribed mean curvature in $${\mathbb {H}}^2\times {\mathbb {R}}$$ H 

@article{Bueno2020ProperlyES,
  title={Properly embedded surfaces with prescribed mean curvature in 
              
                
              
              \$\$\{\mathbb \{H\}\}^2\times \{\mathbb \{R\}\}\$\$
              
                
                  
                    
                      H
                    
                    },
  author={Antonio Bueno},
  journal={Annals of Global Analysis and Geometry},
  year={2020},
  volume={59},
  pages={69-80}
}
  • Antonio Bueno
  • Published 6 October 2020
  • Mathematics
  • Annals of Global Analysis and Geometry
The aim of this paper is to extend classic results of the theory of constant mean curvature surfaces in the product space $${\mathbb {H}}^2\times {\mathbb {R}}$$ H 2 × R to the class of immersed surfaces whose mean curvature is given as a $$C^1$$ C 1 function depending on their angle function. We cover topics such as the existence of a priori curvature and height estimates for graphs and a structure-type result, which classifies properly embedded surfaces with finite topology and at most one… 
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2 Citations

2 Citations

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References

SHOWING 1-10 OF 15 REFERENCES
Uniqueness of immersed spheres in three-manifolds
Let $\mathcal{A}$ be a class of immersed surfaces in a three-manifold $M$, and assume that $\mathcal{A}$ is modeled by an elliptic PDE over each tangent plane. In this paper we solve the so-called…
Height estimates for surfaces with positive constant mean curvature in $\mathbb{M}^{2}\times\mathbb{R}$
We obtain height estimates for compact embedded surfaces with positive constant mean cur- vature in a Riemannian product spaceM 2 £R and boundary on a slice. We prove that these estimates are optimal…
Cylindrically bounded constant mean curvature surfaces in $\mathbb{H}^2\times\mathbb{R}$
In this paper we prove that a properly embedded constant mean curvature surface in $\mathbb{H}^2\times\mathbb{R}$ which has finite topology and stays at a finite distance from a vertical geodesic…
Uniqueness of minimal surfaces whose boundary is a horizontal graph and some Bernstein problems in $${{\mathbb{H}^{2}\times \mathbb{R}}}$$
We deduce that a connected compact immersed minimal surface in $${{\mathbb{H}^{2}\times \mathbb{R}}}$$ whose boundary has an injective horizontal projection on an admissible convex curve in…
Constant mean curvature surfaces in $M^2\times \mathbf{R}$
The subject of this paper is properly embedded H-surfaces in Riemannian three manifolds of the form M 2 x R, where M 2 is a complete Riemannian surface. When M 2 = R 2 , we are in the classical…
Rotational hypersurfaces of prescribed mean curvature
Complete surfaces with positive extrinsic curvature in product spaces
We prove that every complete connected immersed surface with positive extrinsic curvature K in H 2 � R must be properly embedded, homeomorphic to a sphere or a plane and, in the latter case, study…
A characterization of constant mean curvature surfaces in homogeneous 3-manifolds
Isometric immersions into 3-dimensional homogeneous manifolds
We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous Riemannian manifold with a 4-dimensional…
General curvature estimates for stable H-surfaces in 3-manifolds applications
We obtain an estimate for the norm of the second fundamental form of stable H-surfaces in Riemannian 3-manifolds with bounded sectional curvature. Our estimate depends on the distance to the boundary…
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