×
Ma, Bingqing; Huang, Guangyue

Eigenvalue relationships between Laplacians of constant mean curvature hypersurfaces in \(\mathbb{S}^{n+1}\). (English) Zbl 1275.58016

Commun. Math. 21, No. 1, 31-38 (2013).
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.

Cited in 1 Document

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Keywords:

hypersurface with constant mean curvature; the stability operator; Hodge Laplacian; rough Laplacian
PDF BibTeX XML Cite
\textit{B. Ma} and \textit{G. Huang}, Commun. Math. 21, No. 1, 31--38 (2013; Zbl 1275.58016)
Full Text: Link
  • logo of hbz
  • logo of WorldCat

References:

[1] Alencar, H., Carmo, M. do, Colares, A. G.: Stable hypersurfaces with constant scalar curvature. Math. Z., 213, 1993, 117-131, · Zbl 0792.53057
[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, · Zbl 0653.53045
[3] Cao, L., Li, H.: \(r\)-Minimal submanifolds in space forms. Ann. Global Anal. Geom., 32, 2007, 311-341, · Zbl 1168.53029
[4] Cheng, Q.-M.: First eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature. Proc. Amer. Math. Soc., 136, 2008, 3309-3318, · Zbl 1155.53030
[5] Chern, S. S.: Minimal Submanifolds in a Riemannian Manifold (mimeographed). 1968, University of Kansas, Lawrence,
[6] Soufi, A. El, Ilias, S.: Second eigenvalue of Schrödinger operators, mean curvature. Commun. Math. Phys., 208, 2000, 761-770, · Zbl 0955.58025
[7] Li, H.: Hypersurfaces with constant scalar curvature in space forms. Math. Ann., 305, 1996, 665-672, · Zbl 0864.53040
[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
[9] Savo, A.: Index bounds for minimal hypersurfaces of the sphere. Indiana Univ. Math. J., 59, 2010, 823-837, · Zbl 1209.53052
[10] Simons, J.: Minimal varieties in Riemannian manifolds. Ann. of Math., 88, 2, 1968, 62-105, · Zbl 0181.49702
[11] Wu, C.: New characterization of the Clifford tori, the Veronese surface. Arch. Math. (Basel), 61, 1993, 277-284, · Zbl 0791.53056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.