Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

4 Issues per year


IMPACT FACTOR 2017: 0.734

CiteScore 2017: 0.70

SCImago Journal Rank (SJR) 2017: 0.695
Source Normalized Impact per Paper (SNIP) 2017: 0.891

Mathematical Citation Quotient (MCQ) 2017: 0.62

Online
ISSN
1615-7168
See all formats and pricing
More options …
Volume 18, Issue 4

Issues

On the umbilicity of generalized linear Weingarten hypersurfaces in hyperbolic spaces

Cícero P. Aquino / Márcio Batista / Henrique F. de Lima
  • Corresponding author
  • Departamento de Matemática, Universidade Federal de Campina Grande, 58429-970 Campina Grande, Paraíba, Brazil
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-03-26 | DOI: https://doi.org/10.1515/advgeom-2018-0005

Abstract

We deal with complete generalized linear Weingarten hypersurfaces immersed in hyperbolic spaces. Under appropriate constraints on the image of the Gauss map, we present suitable conditions which guarantee the umbilicity of these hypersurfaces.

Keywords: Hyperbolic space; complete two-sided hypersurfaces; totally umbilical hypersurfaces; (r, s)-linear Weingarten hypersurfaces Gauss image

MSC 2010: Primary 53C42; Secondary 53B30; 53C24; 83C99

References

  • [1]

    J. A. Aledo, L. J. Alías, A. Romero, Integral formulas for compact space-like hypersurfaces in de Sitter space: applications to the case of constant higher order mean curvature. J. Geom. Phys. 31 (1999), 195–208. MR1706636 Zbl 0969.53031Google Scholar

  • [2]

    L. J. Alías, A. Brasil, Jr., A. Gervasio Colares, Integral formulae for spacelike hypersurfaces in conformally stationary space-times and applications. Proc. Edinb. Math. Soc. (2) 46 (2003), 465–488. MR1998575 Zbl 1053.53038Google Scholar

  • [3]

    C. C. P. Aquino, H. F. de Lima, M. A. L. Velásquez, A new characterization of complete linear Weingarten hypersurfaces in real space forms. Pacific J. Math. 261 (2013), 33–43. MR3037557 Zbl 1273.53051Google Scholar

  • [4]

    C. C. P. Aquino, H. F. de Lima, M. A. L. Velásquez, Generalized maximum principles and the characterization of linear Weingarten hypersurfaces in space forms. Michigan Math. J. 63 (2014), 27–40. MR3189466 Zbl 1304.53051Google Scholar

  • [5]

    C. C. P. Aquino, H. F. de Lima, M. A. L. Velásquez, Linear Weingarten hypersurfaces with bounded mean curvature in the hyperbolic space. Glasg. Math. J. 57 (2015), 653–663. MR3395339 Zbl 1327.53075Google Scholar

  • [6]

    C. C. P. Aquino, H. F. de Lima, On the geometry of linear Weingarten hypersurfaces in the hyperbolic space. Monatsh. Math. 171 (2013), 259–268. MR3090789 Zbl 1279.53055Google Scholar

  • [7]

    J. A. L. M. Barbosa, A. G. Colares, Stability of hypersurfaces with constant r-mean curvature. Ann. Global Anal. Geom. 15 (1997), 277–297. MR1456513 Zbl 0891.53044Google Scholar

  • [8]

    A. Barros, J. Silva, P. Sousa, Rotational linear Weingarten surfaces into the Euclidean sphere. Israel J. Math. 192 (2012), 819–830. MR3009743 Zbl 1259.53050Google Scholar

  • [9]

    A. Caminha, On spacelike hypersurfaces of constant sectional curvature Lorentz manifolds. J. Geom. Phys. 56 (2006), 1144–1174. MR2234043 Zbl 1102.53044Google Scholar

  • [10]

    A. Caminha, The geometry of closed conformal vector fields on Riemannian spaces. Bull. Braz. Math. Soc. (N.S.) 42 (2011), 277–300. MR2833803 Zbl 1242.53068Google Scholar

  • [11]

    X. Chao, Y. Lv, On the Gauss map of Weingarten hypersurfaces in hyperbolic spaces. Bull. Braz. Math. Soc. (N.S.) 47 (2016), 1051–1069. MR3582027 Zbl 1369.53040Google Scholar

  • [12]

    H. Chen, X. Wang, Stability and eigenvalue estimates of linear Weingarten hypersurfaces in a sphere. J. Math. Anal. Appl. 397 (2013), 658–670. MR2979602 Zbl 1260.53112Google Scholar

  • [13]

    H. Li, Y. J. Suh, G. Wei, Linear Weingarten hypersurfaces in a unit sphere. Bull. Korean Math. Soc. 46 (2009), 321–329. MR2502796 Zbl 1165.53361Google Scholar

  • [14]

    S. Montiel, Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces. J. Math. Soc. Japan 55 (2003), 915–938. MR2003752 Zbl 1049.53044Google Scholar

  • [15]

    R. C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differential Geometry 8 (1973), 465–477. MR0341351 Zbl 0277.53030Google Scholar

  • [16]

    H. Rosenberg, Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117 (1993), 211–239. MR1216008 Zbl 0787.53046Google Scholar

  • [17]

    S. Shu, Linear Weingarten hypersurfaces in a real space form. Glasg. Math. J. 52 (2010), 635–648. MR2679920 Zbl 1203.53059Google Scholar

  • [18]

    M. A. Velásquez, A. F. de Sousa, H. F. de Lima, On the stability of hypersurfaces in space forms. J. Math. Anal. Appl. 406 (2013), 134–146. MR3062407 Zbl 1309.53050Google Scholar

  • [19]

    S. T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry. Indiana Univ. Math. J. 25 (1976), 659–670. MR0417452 Zbl 0335.53041Google Scholar

About the article


Received: 2016-07-15

Published Online: 2018-03-26

Published in Print: 2018-10-25


Funding: The first author is partially supported by CNPq, Brazil, grant 302738/2014-2. The second author is partially supported by CNPq, Brazil, grant 456755/2014-4. The third author is partially supported by CNPq, Brazil, grant 303977/2015-9.


Citation Information: Advances in Geometry, Volume 18, Issue 4, Pages 425–430, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0005.

Export Citation

© 2018 Walter de Gruyter GmbH Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in