Czechoslovak Mathematical Journal, first online, pp. 1-13


Sharp eigenvalue estimates of closed $H$-hypersurfaces in locally symmetric spaces

Eudes L. de Lima, Henrique F. de Lima, Fábio R. dos Santos, Marco A. L. Velásquez

Received December 10, 2017.   Published online March 28, 2019.

Abstract:  The purpose of this article is to obtain sharp estimates for the first eigenvalue of the stability operator of constant mean curvature closed hypersurfaces immersed into locally symmetric Riemannian spaces satisfying suitable curvature conditions (which includes, in particular, a Riemannian space with constant sectional curvature). As an application, we derive a nonexistence result concerning strongly stable hypersurfaces in these ambient spaces.
Keywords:  locally symmetric Riemannian space; closed $H$-hypersurface; strong stability; first stability eigenvalue
Classification MSC:  53C42, 53A10
DOI:  10.21136/CMJ.2019.0562-17

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] H. Alencar, M. P. do Carmo: Hypersurfaces with constant mean curvature in spheres. Proc. Am. Math. Soc. 120 (1994), 1223-1229. DOI 10.2307/2160241 | MR 1172943 | Zbl 0802.53017
[2] L. J. Alías, A. Barros, A. Brasil Jr.: A spectral characterization of the $H(r)$-torus by the first stability eigenvalue. Proc. Am. Math. Soc. 133 (2005), 875-884. DOI 10.1090/S0002-9939-04-07559-8 | MR 2113939 | Zbl 1065.53046
[3] L. J. Alías, A. Brasil Jr., O. Perdomo: On the stability index of hypersurfaces with constant mean curvature in spheres. Proc. Am. Math. Soc. 135 (2007), 3685-3693. DOI 10.1090/S0002-9939-07-08886-7 | MR 2336585 | Zbl 1157.53030
[4] L. J. Alías, H. F. de Lima, J. Meléndez, F. R. dos Santos: Rigidity of linear Weingarten hypersurfaces in locally symmetric manifolds. Math. Nachr. 289 (2016), 1309-1324. DOI 10.1002/mana.201400296 | MR 3541811 | Zbl 1350.53078
[5] L. J. Alías, T. Kurose, G. Solanes: Hadamard-type theorems for hypersurfaces in hyperbolic spaces. Differ. Geom. Appl. 24 (2006), 492-502. DOI 10.1016/j.difgeo.2006.02.008 | MR 2254052 | Zbl 1103.52006
[6] L. J. Alías, M. A. Meroño, I. Ortiz: On the first stability eigenvalue of constant mean curvature surfaces into homogeneous 3-manifolds. Mediterr. J. Math. 12 (2015), 147-158. DOI 10.1007/s00009-014-0397-y | MR 3306032 | Zbl 1316.53062
[7] A. A. de Barros, A. C. Brasil Jr., L. A. M. de Sousa Jr.: A new characterization of submanifolds with parallel mean curvature vector in $\mathbb{S}^{n + p}$. Kodai Math. J. 27 (2004), 45-56. DOI 10.2996/kmj/1085143788 | MR 2042790 | Zbl 1059.53047
[8] J. N. Gomes, H. F. de Lima, F. R. dos Santos, M. A. L. Velásquez: Complete hypersurfaces with two distinct principal curvatures in a locally symmetric Riemannian manifold. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 133 (2016), 15-27. DOI 10.1016/j.na.2015.11.026 | MR 3449745 | Zbl 1333.53090
[9] J. Melendéz: Rigidity theorems for hypersurfaces with constant mean curvature. Bull. Braz. Math. Soc. 45 (2014), 385-404. DOI 10.1007/s00574-014-0055-9 | MR 3264798 | Zbl 1319.53065
[10] M. A. Meroño, I. Ortiz: Eigenvalue estimates for the stability operator of CMC compact surfaces in three-dimensional warped products. J. Math. Anal. Appl. 434 (2016), 1779-1788. DOI 10.1016/j.jmaa.2015.10.016 | MR 3415751 | Zbl 1328.53075
[11] M. A. Meroño, I. Ortiz: On the first stability eigenvalue of CMC surfaces into warped products with two-dimensional fiber. Differ. Geom. Appl. 45 (2016), 67-77. DOI 10.1016/j.difgeo.2015.11.009 | MR 3457388 | Zbl 1334.53061
[12] M. Okumura: Hypersurfaces and a pinching problem on the second fundamental tensor. Am. J. Math. 96 (1974), 207-213. DOI 10.2307/2373587 | MR 0353216 | Zbl 0302.53028
[13] O. Perdomo,: First stability eigenvalue characterization of Clifford hypersurfaces. Proc. Am. Math. Soc. 130 (2002), 3379-3384. DOI 10.1090/S0002-9939-02-06451-1 | MR 1913017 | Zbl 1014.53036
[14] J. Simons: Minimal varietes in Riemannian manifolds. Ann. Math. 88 (1968), 62-105. DOI 10.2307/1970556 | MR 0233295 | Zbl 0181.49702
[15] M. A. L. Velásquez, H. F. de Lima, F. R. dos Santos, C. P. Aquino: On the first stability eigenvalue of hypersurfaces in the Euclidean and hyperbolic spaces. Quaest. Math. 40 (2017), 605-616. DOI 10.2989/16073606.2017.1305463 | MR 3691472
[16] C. Wu: New characterization of the Clifford tori and the Veronese surface. Arch. Math. 61 (1993), 277-284. DOI 10.1007/BF01198725 | MR 1231163 | Zbl 0791.53056

Affiliations:   Eudes L. de Lima, Unidade Acadêmica de Ciências Exatas e da Natureza, Universidade Federal de Campina Grande, 58.900-000 Cajazeiras, Paraíba, Brazil, e-mail: eudes.lima@ufcg.edu.br; Henrique F. de Lima (corresponding author), Fábio R. dos Santos, Marco A. L. Velásquez, Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil, e-mail: henrique@mat.ufcg.edu.br, fabio@mat.ufcg.edu.br, marco.velasquez@mat.ufcg.edu.br


 
PDF available at: