A characterization of constant mean curvature surfaces in homogeneous 3-manifolds

By I. Fernández, Pablo Mira. 2005

61 citations recorded by the source corpus.

Abstract It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this paper we describe all the surfaces with holomorphic Hopf differential in the homogeneous 3-manifolds isometric to H 2 × R or having isometry group isomorphic either to the one of the universal cover of PSL ( 2 , R ) , or to the one of a certain class of Berger spheres. It turns out that, except for the case of these Berger spheres, there exist some exceptional surfaces with holomorphic Hopf differential and non-constant mean curvature.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/396c31dab5ee3f30503639bd1efc2b07f2f71923

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Harmonic maps and constant mean curvature surfaces in H 2 x R (2007, background): “Recently, Abresch [AbRo2] (see also [BeTa, FeMi ]) has announced the existence of a holomorphic quadratic differential for CMC surfaces in the Heisenberg group Nil3.”
Minimal translation surfaces in the Heisenberg group Nil3 (2011, background): “We refer the survey [1],[7] or lecture notes [4] and references therein.”
Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space (2007, background): “[AbRo2, Dan1, Dan2, FeMi1, FeMi2 ] and references therein.”
PARABOLIC AND HYPERBOLIC SCREW MOTION SURFACES IN ℍ2×ℝ (2008, background): “For other relevant papers on this subject the reader is referred to William Meeks III and Harold Rosenberg [13], Barbara Nelli and Harold Rosenberg [14], Isabel Fernández and Pablo Mira [8] .”