| Title:
|
Sharp eigenvalue estimates of closed $H$-hypersurfaces in locally symmetric spaces (English) |
| Author:
|
de Lima, Eudes L. |
| Author:
|
de Lima, Henrique F. |
| Author:
|
dos Santos, Fábio R. |
| Author:
|
Velásquez, Marco A. L. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
4 |
| Year:
|
2019 |
| Pages:
|
969-981 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
locally symmetric Riemannian space |
| Keyword:
|
closed $H$-hypersurface |
| Keyword:
|
strong stability |
| Keyword:
|
first stability eigenvalue |
| MSC:
|
53A10 |
| MSC:
|
53C42 |
| idZBL:
|
07144868 |
| idMR:
|
MR4039613 |
| DOI:
|
10.21136/CMJ.2019.0562-17 |
| . |
| Date available:
|
2019-11-28T08:47:58Z |
| Last updated:
|
2022-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147907 |
| . |
| Reference:
|
[1] Alencar, H., Carmo, M. P. do: Hypersurfaces with constant mean curvature in spheres.Proc. Am. Math. Soc. 120 (1994), 1223-1229. Zbl 0802.53017, MR 1172943, 10.2307/2160241 |
| Reference:
|
[2] Alías, L. J., Barros, A., Jr., A. Brasil: A spectral characterization of the $H(r)$-torus by the first stability eigenvalue.Proc. Am. Math. Soc. 133 (2005), 875-884. Zbl 1065.53046, MR 2113939, 10.1090/S0002-9939-04-07559-8 |
| Reference:
|
[3] Alías, L. J., Jr., A. Brasil, Perdomo, O.: On the stability index of hypersurfaces with constant mean curvature in spheres.Proc. Am. Math. Soc. 135 (2007), 3685-3693. Zbl 1157.53030, MR 2336585, 10.1090/S0002-9939-07-08886-7 |
| Reference:
|
[4] Alías, L. J., Lima, H. F. de, Meléndez, J., Santos, F. R. dos: Rigidity of linear Weingarten hypersurfaces in locally symmetric manifolds.Math. Nachr. 289 (2016), 1309-1324. Zbl 1350.53078, MR 3541811, 10.1002/mana.201400296 |
| Reference:
|
[5] Alías, L. J., Kurose, T., Solanes, G.: Hadamard-type theorems for hypersurfaces in hyperbolic spaces.Differ. Geom. Appl. 24 (2006), 492-502. Zbl 1103.52006, MR 2254052, 10.1016/j.difgeo.2006.02.008 |
| Reference:
|
[6] Alías, L. J., Meroño, M. A., Ortiz, I.: On the first stability eigenvalue of constant mean curvature surfaces into homogeneous 3-manifolds.Mediterr. J. Math. 12 (2015), 147-158. Zbl 1316.53062, MR 3306032, 10.1007/s00009-014-0397-y |
| Reference:
|
[7] Barros, A. A. de, Jr., A. C. Brasil, Jr., L. A. M. de Sousa: A new characterization of submanifolds with parallel mean curvature vector in $\mathbb{S}^{n + p}$.Kodai Math. J. 27 (2004), 45-56. Zbl 1059.53047, MR 2042790, 10.2996/kmj/1085143788 |
| Reference:
|
[8] Gomes, J. N., Lima, H. F. de, Santos, F. R. dos, Velásquez, M. A. L.: 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. Zbl 1333.53090, MR 3449745, 10.1016/j.na.2015.11.026 |
| Reference:
|
[9] Melendéz, J.: Rigidity theorems for hypersurfaces with constant mean curvature.Bull. Braz. Math. Soc. 45 (2014), 385-404. Zbl 1319.53065, MR 3264798, 10.1007/s00574-014-0055-9 |
| Reference:
|
[10] Meroño, M. A., Ortiz, I.: Eigenvalue estimates for the stability operator of CMC compact surfaces in three-dimensional warped products.J. Math. Anal. Appl. 434 (2016), 1779-1788. Zbl 1328.53075, MR 3415751, 10.1016/j.jmaa.2015.10.016 |
| Reference:
|
[11] Meroño, M. A., Ortiz, I.: On the first stability eigenvalue of CMC surfaces into warped products with two-dimensional fiber.Differ. Geom. Appl. 45 (2016), 67-77. Zbl 1334.53061, MR 3457388, 10.1016/j.difgeo.2015.11.009 |
| Reference:
|
[12] Okumura, M.: Hypersurfaces and a pinching problem on the second fundamental tensor.Am. J. Math. 96 (1974), 207-213. Zbl 0302.53028, MR 0353216, 10.2307/2373587 |
| Reference:
|
[13] Perdomo,, O.: First stability eigenvalue characterization of Clifford hypersurfaces.Proc. Am. Math. Soc. 130 (2002), 3379-3384. Zbl 1014.53036, MR 1913017, 10.1090/S0002-9939-02-06451-1 |
| Reference:
|
[14] Simons, J.: Minimal varietes in Riemannian manifolds.Ann. Math. 88 (1968), 62-105. Zbl 0181.49702, MR 0233295, 10.2307/1970556 |
| Reference:
|
[15] Velásquez, M. A. L., Lima, H. F. de, Santos, F. R. dos, Aquino, C. P.: On the first stability eigenvalue of hypersurfaces in the Euclidean and hyperbolic spaces.Quaest. Math. 40 (2017), 605-616. MR 3691472, 10.2989/16073606.2017.1305463 |
| Reference:
|
[16] Wu, C.: New characterization of the Clifford tori and the Veronese surface.Arch. Math. 61 (1993), 277-284. Zbl 0791.53056, MR 1231163, 10.1007/BF01198725 |
| . |
