Abstract
We prove that any complete surface with constant mean curvature in a homogeneous space \(\mathbb{E}(\kappa,\tau)\) which is transversal to the vertical Killing vector field is, in fact, a vertical graph. As a consequence we get that any orientable, parabolic, complete, immersed surface with constant mean curvature H in \(\mathbb{E}(\kappa,\tau)\) (different from a horizontal slice in \(\mathbb{S}^{2}\times\mathbb{R}\)) is either a vertical cylinder or a vertical graph (in both cases, it must be 4H 2+κ≤0).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barbosa, J.L., do Carmo, M., Eschenburg, J.: Stability of hypersurfaces with constant mean curvature in Riemannian manifolds. Math. Z. 197, 123–138 (1988)
Collin, P., Rosenberg, H.: Construction of harmonic diffeomorphisms and minimal graphs. Ann. Math. 172, 1879–1906 (2010)
Daniel, B.: Isometric immersions into 3-dimensional homogeneous manifolds. Comment. Math. Helv. 82(1), 87–131 (2007)
Daniel, B., Hauswirth, L.: Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group. Proc. Lond. Math. Soc. (3) 98(2), 445–470 (2009)
Daniel, B., Hauswirth, L., Mira, P.: Lecture notes on homogeneous 3-manifolds. In: 4th KIAS Workshop on Differential Geometry, Korea Institute for Advanced Study, Seoul, Korea (2009)
Espinar, J.M., Rosenberg, H.: Complete constant mean curvature surfaces and Bernstein type theorems in M 2×R. J. Differ. Geom. 82(3), 611–628 (2009)
Fernández, I., Mira, P.: Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space. Transl. Am. Math. Soc. 361, 5737–5752 (2009)
Fischer-Colbrie, D.: On complete minimal surfaces with finite Morse index in 3-manifolds. Invent. Math. 82, 121–132 (1985)
Folha, A., Melo, S.: The Dirichlet problem for constant mean curvature graphs in \(\mathbb{H}^{2}\times\mathbb{R}\) over unbounded domains. Pac. J. Math. 251(1), 37–65 (2011)
Hauswirth, L., Rosenberg, H., Spruck, J.: On complete mean curvature \(\frac{1}{2}\) surfaces in \(\mathbb{H} ^{2}\times\mathbb{R}\). Commun. Anal. Geom. 16, 989–1005 (2008)
Hauswirth, L., Rosenberg, H., Spruck, J.: Infinite boundary value problems for constant mean curvature graphs in \(\mathbb{H}\times\mathbb{R}\) and \(\mathbb{S}\times \mathbb{R}\). Am. J. Math. 131, 195–226 (2009)
Manzano, J.M.: On the classification of Killing submersions and their isometries. Preprint, available at arXiv:1211.2115 [math.DG]
Manzano, J.M., Pérez, J., Rodríguez, M.M.: Parabolic stable surfaces with constant mean curvature. Calc. Var. Partial Differ. Equ. 42(1–2), 137–152 (2011)
Mazet, L., Rodríguez, M.M., Rosenberg, H.: The Dirichlet problem for the minimal surface equation—with possible infinite boundary data—over domains in a Riemannian surface. Proc. Lond. Math. Soc. 102(3), 985–1023 (2011)
Meeks, W.H. III, Pérez, J., Ros, A.: Stable constant mean curvature surfaces. In: Ji, L., Li, P., Schoen, R., Simon, L. (eds.) Handbook of Geometrical Analysis, vol. 1, pp. 301–380. International Press, Somerville (2008)
Melo, S.: Minimal graphs in PSL\(_{2}(\mathbb{R})\) over unbounded domains. Preprint
Nelli, B., Rosenberg, H.: Minimal surfaces in \(\mathbb{H}^{2}\times\mathbb{R}\). Bull. Braz. Math. Soc. 33(2), 263–292 (2002)
Nelli, B., Rosenberg, H.: Global properties of constant mean curvature surfaces in \(\mathbb{H} ^{2}\times\mathbb{R}\). Pac. J. Math. 226(1), 137–152 (2006)
Peñafiel, C.: Graphs and multi-graphs in homogeneous 3-manifolds with isometry groups of dimension 4. Proc. Am. Math. Soc. 140(7), 2465–2478 (2012)
Rosenberg, H.: Constant mean curvature surfaces in homogeneously regular 3-manifolds. Bull. Aust. Math. Soc. 74, 227–238 (2006)
Sa Earp, R.: Parabolic and hyperbolic screw motion surfaces in \(\mathbb{H} ^{2}\times\mathbb{R}\). Bull. Aust. Math. Soc. 85, 113–143 (2008)
Younes, R.: Minimal surfaces in \(\widetilde{\mathrm{PSL}_{2}}(\mathbb{R})\). Ill. J. Math. 54(2), 671–712 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by the MCyT-Feder research project MTM2011-22547, the Regional J. Andalucía Grant no. P09-FQM-5088 and the CEI BioTIC GENIL project (CEB09-0010) no. PYR-2010-21.
Rights and permissions
About this article
Cite this article
Manzano, J.M., Rodríguez, M.M. On Complete Constant Mean Curvature Vertical Multigraphs in \(\mathbb{E}(\kappa,\tau)\) . J Geom Anal 25, 336–346 (2015). https://doi.org/10.1007/s12220-013-9431-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-013-9431-8