A Hopf differential for constant mean curvature surfaces inS2×R andH2×R

By U. Abresch, H. Rosenberg. 2004

235 citations recorded by the source corpus.

A basic tool in the theory of constant mean curvature (cmc) surfaces Σ in space forms is the holomorphic quadratic differential discovered by H. Hopf. In this paper we generalize this differential to immersed cmc surfaces Σ in the product spaces S × R and H × R and prove a corresponding result about the geometry of immersed cmc spheres S in these target spaces.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9f4f0db1e465b6698bb9bf3bdef86e7922ca3381

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Complete surfaces with positive extrinsic curvature in product spaces (2007, background): “Rosenberg [AR1] defined a holomorphic quadratic differential on constant mean curvature surfaces in the homogeneous 3-manifolds.”
Surfaces in Three-Dimensional Lie Groups (2005, methodology): “We also hope that our approach will be helpful in studying global properties of minimal and constant mean curvature surfaces in Lie groups in the spirit of [ 2 , 4].”
Isometric immersions into S^n x R and H^n x R and applications to minimal surfaces (2004, background): “MR1757077 (2001k:53120) [Ros02a] H. Rosenberg, Bryant surfaces, The global theory of minimal surfaces in flat spaces (Martina Franca, 1999), Lecture Notes in Math., vol. 1775, Springer, Berlin, 2002, pp. 67–111.”
The Gauss Map of Minimal Surfaces in the Heisenberg Group (2006, background): “On the other hand, U. Abresch and H. Rosenberg proved the existence of a holomorphic quadratic differential for CMC surfaces in Nil3, and more generally in other homogeneous 3-manifolds ([ AR04 ], [AR05]); this quadratic differential is a complex linear combination of the (2,0) part of the second fu”
Associate and conjugate minimal immersions in MxR (2005, background): “For related works on minimal surfaces in M × R , see for instance Daniel [4], Nelli and Rosenberg [12], Meeks and Rosenberg [11] and Rosenberg [13].”
GLOBAL PROPERTIES OF CONSTANT MEAN CURVATURE SURFACES IN H 2 ◊R (2006, background): “This is contained in [Abresch and Rosenberg 2004], where the authors prove that any immersed constant-mean-curvature sphere is rotational.”
Complete surfaces of constant curvature in H2 × R and S2 × R (2005, background): “Lately, theHopf theorem on the characterization of the totally umbilical spheres as the unique topological spheres of constant mean curvature immersed into a 3-dimensional space form, has been extended to the homogeneous spaces H2 × R and S2 ×R by Abresch and Rosenberg [1].”
Totally umbilic surfaces in homogeneous 3-manifolds (2006, unknown): “See for example [14], [6], [7], [17], [15], [12], [13], [11], [9], [4], [2], [10], [5] and [8].”
Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds (2009, background): “Abresch and Rosenberg [1] proved that a complete constant mean curvature surface in S2 ×R and H2 ×R with vanishing Abresch–Rosenberg differential must be rotationally invariant (that is, invariant under a 1-parameter group of isometries acting trivially on the fiber).”