Hypersurfaces of constant higher order mean curvature in warped products
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- by Luis J. Alías, Debora Impera and Marco Rigoli PDF
- Trans. Amer. Math. Soc. 365 (2013), 591-621 Request permission
Abstract:
In this paper we characterize compact and complete hypersurfaces with some constant higher order mean curvature into warped product spaces. Our approach is based on the use of a new trace operator version of the Omori-Yau maximum principle which seems to be interesting in its own.References
- A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl. (4) 58 (1962), 303–315. MR 143162, DOI 10.1007/BF02413056
- Luis J. Alías and A. Gervasio Colares, Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes, Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 3, 703–729. MR 2373968, DOI 10.1017/S0305004107000576
- Luis J. Alías and Marcos Dajczer, Uniqueness of constant mean curvature surfaces properly immersed in a slab, Comment. Math. Helv. 81 (2006), no. 3, 653–663. MR 2250858, DOI 10.4171/CMH/68
- Luis J. Alías and Marcos Dajczer, Constant mean curvature hypersurfaces in warped product spaces, Proc. Edinb. Math. Soc. (2) 50 (2007), no. 3, 511–526. MR 2360513, DOI 10.1017/S0013091505001069
- Luis J. Alías, Jorge H. S. de Lira, and J. Miguel Malacarne, Constant higher-order mean curvature hypersurfaces in Riemannian spaces, J. Inst. Math. Jussieu 5 (2006), no. 4, 527–562. MR 2261223, DOI 10.1017/S1474748006000077
- Luis J. Alías, Debora Impera, and Marco Rigoli, Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson-Walker spacetimes, Math. Proc. Cambridge Philos. Soc. 152 (2012), no. 2, 365–383. MR 2887879, DOI 10.1017/S0305004111000697
- João Lucas Marques Barbosa and Antônio Gervasio Colares, Stability of hypersurfaces with constant $r$-mean curvature, Ann. Global Anal. Geom. 15 (1997), no. 3, 277–297. MR 1456513, DOI 10.1023/A:1006514303828
- Maria Fernanda Elbert, Constant positive 2-mean curvature hypersurfaces, Illinois J. Math. 46 (2002), no. 1, 247–267. MR 1936088
- Lars Gȧrding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957–965. MR 0113978, DOI 10.1512/iumj.1959.8.58061
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Sebastián Montiel, Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds, Indiana Univ. Math. J. 48 (1999), no. 2, 711–748. MR 1722814, DOI 10.1512/iumj.1999.48.1562
- Hideki Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 205–214. MR 215259, DOI 10.2969/jmsj/01920205
- Stefano Pigola, Marco Rigoli, and Alberto G. Setti, A Liouville-type result for quasi-linear elliptic equations on complete Riemannian manifolds, J. Funct. Anal. 219 (2005), no. 2, 400–432. MR 2109258, DOI 10.1016/j.jfa.2004.05.009
- Stefano Pigola, Marco Rigoli, and Alberto G. Setti, Maximum principles on Riemannian manifolds and applications, Mem. Amer. Math. Soc. 174 (2005), no. 822, x+99. MR 2116555, DOI 10.1090/memo/0822
- Shing Tung Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. MR 431040, DOI 10.1002/cpa.3160280203
Additional Information
- Luis J. Alías
- Affiliation: Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain
- Email: ljalias@um.es
- Debora Impera
- Affiliation: Dipartimento di Matematica, Università degli studi di Milano, via Saldini 50, I-20133 Milano, Italy
- Email: debora.impera@unimi.it
- Marco Rigoli
- Affiliation: Dipartimento di Matematica, Università degli studi di Milano, via Saldini 50, I-20133 Milano, Italy
- MR Author ID: 148315
- Email: marco.rigoli@unimi.it
- Received by editor(s): January 11, 2011
- Published electronically: September 12, 2012
- Additional Notes: The first author was partially supported by MICINN project MTM2009-10418 and Fundación Séneca project 04540/GERM/06, Spain. This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Regional Agency for Science and Technology (Regional Plan for Science and Technology 2007-2010).
The third author was partially supported by MEC Grant SAB2010-0073 - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 591-621
- MSC (2010): Primary 53C40, 53C42, 53A10
- DOI: https://doi.org/10.1090/S0002-9947-2012-05774-6
- MathSciNet review: 2995367