| Title:
|
Eigenvalue relationships between Laplacians of constant mean curvature hypersurfaces in $\mathbb{S}^{n+1}$ (English) |
| Author:
|
Ma, Bingqing |
| Author:
|
Huang, Guangyue |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
21 |
| Issue:
|
1 |
| Year:
|
2013 |
| Pages:
|
31-38 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For compact hypersurfaces with constant mean curvature in the unit sphere, we give a comparison theorem between eigenvalues of the stability operator and that of the Hodge Laplacian on 1-forms. Furthermore, we also establish a comparison theorem between eigenvalues of the stability operator and that of the rough Laplacian. (English) |
| Keyword:
|
hypersurface with constant mean curvature |
| Keyword:
|
the stability operator |
| Keyword:
|
Hodge Laplacian |
| Keyword:
|
rough Laplacian |
| MSC:
|
58J50 |
| idZBL:
|
Zbl 06202723 |
| idMR:
|
MR3067120 |
| . |
| Date available:
|
2013-07-18T15:25:33Z |
| Last updated:
|
2014-07-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143347 |
| . |
| Reference:
|
[1] Alencar, H., Carmo, M. do, Colares, A. G.: Stable hypersurfaces with constant scalar curvature.Math. Z., 213, 1993, 117-131, Zbl 0792.53057, MR 1217674, 10.1007/BF03025712 |
| Reference:
|
[2] Barbosa, J. L., Carmo, M. do, Eschenburg, M.: Stability of hypersurfaces with constant mean curvature in Riemannian manifolds.Math. Z., 197, 1988, 123-138, MR 0917854, 10.1007/BF01161634 |
| Reference:
|
[3] Cao, L., Li, H.: $r$-Minimal submanifolds in space forms..Ann. Global Anal. Geom., 32, 2007, 311-341, Zbl 1168.53029, MR 2346221, 10.1007/s10455-007-9064-x |
| Reference:
|
[4] Cheng, Q.-M.: First eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature.Proc. Amer. Math. Soc., 136, 2008, 3309-3318, MR 2407097, 10.1090/S0002-9939-08-09304-0 |
| Reference:
|
[5] Chern, S. S.: Minimal Submanifolds in a Riemannian Manifold (mimeographed).1968, University of Kansas, Lawrence, MR 0248648 |
| Reference:
|
[6] Soufi, A. El, Ilias, S.: Second eigenvalue of Schrödinger operators, mean curvature.Commun. Math. Phys., 208, 2000, 761-770, MR 1736334, 10.1007/s002200050009 |
| Reference:
|
[7] Li, H.: Hypersurfaces with constant scalar curvature in space forms.Math. Ann., 305, 1996, 665-672, Zbl 0864.53040, MR 1399710, 10.1007/BF01444243 |
| Reference:
|
[8] Li, H., Wang, X.: Second eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature.Proc. Amer. Math. Soc., 140, 2012, 291-307, Zbl 1252.53071, MR 2833541, 10.1090/S0002-9939-2011-10892-X |
| Reference:
|
[9] Savo, A.: Index bounds for minimal hypersurfaces of the sphere.Indiana Univ. Math. J., 59, 2010, 823-837, Zbl 1209.53052, MR 2779062, 10.1512/iumj.2010.59.4013 |
| Reference:
|
[10] Simons, J.: Minimal varieties in Riemannian manifolds.Ann. of Math., 88, 2, 1968, 62-105, Zbl 0181.49702, MR 0233295, 10.2307/1970556 |
| Reference:
|
[11] Wu, C.: New characterization of the Clifford tori, the Veronese surface.Arch. Math. (Basel), 61, 1993, 277-284, MR 1231163, 10.1007/BF01198725 |
| . |
