Graphs and multi-graphs in homogeneous 3-manifolds with isometry groups of dimension 4

Author:
Carlos Peñafiel

Journal:
Proc. Amer. Math. Soc. **140** (2012), 2465-2478

MSC (2000):
Primary 53A35

DOI:
https://doi.org/10.1090/S0002-9939-2011-11075-X

Published electronically:
October 25, 2011

MathSciNet review:
2898709

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the existence of multi-graphs which are immersed in , having constant mean curvature , where is a homogeneous, simply connected 3-manifold whose isometry group has dimension 4.

**1.**B. Daniel, Isometric Immersions into 3-Dimensional Homogeneous Manifolds. Comment. Math. Helv., 82 (2007), no. 1, 87-131. MR**2296059 (2008a:53058)****2.**C. Peñafiel, Surfaces of constant mean curvature in homogeneous three manifolds with emphasis in , Doctoral Thesis - PUC-RJ, Brazil, 2010.**3.**E. Toubiana, Note sur les Varietes Homogenes de Dimension 3. Preprint.**4.**H. Rosenberg, R. Souam, E. Toubiana, General Curvature Estimates for Stable -Surfaces in 3-Manifolds and Applications. J. Differential Geom. 84 (2010), no. 3, 623-648. MR**2669367****5.**I. Salavessa, Graphs with parallel mean curvature. Proc. Amer. Math. Soc. 107 (1989), no. 2, 449-458. MR**965247 (90a:53072)****6.**J. Barbosa, M. do Carmo, Stability of hypersurfaces with constant mean curvature. Math. Z., 185(3) (1984). MR**731682 (85k:58021c)****7.**J. Barbosa, G. Bessa, J. Montenegro, On Bernstein-Heinz-Chern-Flanders Inequalities. Mathematical Proceedings of the Cambridge Philosophical Society, v. 144, pp. 457-464, 2008. MR**2405902 (2009c:53039)****8.**J. Cheeger, D. Ebin, Comparison Theorems in Riemannian Geometry. North-Holland Publishing Company, 1975. MR**0458335 (56:16538)****9.**J. Espinar, H. Rosenberg, Complete Constant Mean Curvature Surfaces in Homogeneous Spaces. To appear in Comment. Math. Helv., 2010.**10.**J. Espinar, H. Rosenberg, Complete Constant Mean Curvature Surfaces and Bernstein Type Theorems in . J. Differential Geom. 82 (2009), no. 3, 611-628. MR**2534989 (2010m:53015)****11.**L. Hauswirth, H. Rosenberg, J. Spruck, On Complete Mean Curvature 1/2 Surfaces in . Comm. Anal. Geom. 16 (5) (2009), 989-1005. MR**2471365 (2010d:53009)****12.**M. do Carmo, J. Barbosa, J. Eschenburg, Stability of Hypersurfaces of Riemannian Manifolds with Constant Mean Curvature, Math. Z. 197 (1988), 123-138. MR**917854 (88m:53109)****13.**R. Sa Earp, Parabolic and Hyperbolic Screw Motion Surfaces in and , J. Aust. Math. Soc. (2008), 113-143. MR**2460869 (2010d:53067)****14.**W. Thurston. Three-Dimensional Geometry and Topology. Princeton, 1997. MR**1435975 (97m:57016)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
53A35

Retrieve articles in all journals with MSC (2000): 53A35

Additional Information

**Carlos Peñafiel**

Affiliation:
Instituto de Matemática, Universidade Federal de Rio de Janeiro, Rio de Janeiro, 22453-900, Brazil

Email:
penafiel@im.ufrj.br

DOI:
https://doi.org/10.1090/S0002-9939-2011-11075-X

Keywords:
Constant mean curvature,
graphs and multi-graphs,
homogeneous 3-manifolds

Received by editor(s):
July 29, 2010

Received by editor(s) in revised form:
February 14, 2011

Published electronically:
October 25, 2011

Additional Notes:
The author was supported by FAPER, Brazil

Communicated by:
Jianguo Cao

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.