Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T13:17:02.541Z Has data issue: false hasContentIssue false
  • English
  • Français

Eigenvalue estimates for submanifolds of warped product spaces

Published online by Cambridge University Press:  28 June 2013

G. P. BESSA ,
S. C. GARCÍA–MARTÍNEZ ,
L. MARI  and
H. F. RAMIREZ–OSPINA
G. P. BESSA
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, 60455-760 Fortaleza-CE, Brazil. e-mail: [email protected], [email protected]
S. C. GARCÍA–MARTÍNEZ
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain. e-mail: [email protected], [email protected]
L. MARI
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, 60455-760 Fortaleza-CE, Brazil. e-mail: [email protected], [email protected]
H. F. RAMIREZ–OSPINA
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain. e-mail: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we give lower bounds for the fundamental tone of open sets in minimal submanifolds immersed into warped product spaces of type Nn ×f Qq, where f ∈ C∞(N). This setting allows us to deal, among other things, with minimal submanifolds bounded by cylinders, cones, spheres and pseudo-hyperbolic spaces where most of these examples are not covered in the literature. Applications also include the study of the essential spectrum of hyperbolic graphs over compact regions of the boundary at infinity.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 1 , January 2014 , pp. 25 - 42
Copyright
Copyright © Cambridge Philosophical Society 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Barbosa, J. L. and Do Carmo, M. P.Stability of minimal surfaces and eigenvalues of the Laplacian. Math. Z. 173 (1980), 13–28.CrossRefGoogle Scholar
[2]Barroso, C. S. and Bessa, G. P.A note on the first eigenvalue of spherically symmetric manifolds. Mat. Contemp. 30 (2006), 63–69.Google Scholar
[3]Barroso, C. S. and Bessa, G. P.Lower bounds for the first Laplacian eigenvalue of geodesic balls of spherically symmetric manifolds. Int. J. Appl. Math. Stat. 6 (2006), 82–86.Google Scholar
[4]Barta, J.Sur la vibration fundamentale d'une membrane. C. R. Acad. Sci. 204 (1937), 472–473.Google Scholar
[5]Bérard, P. H.Spectral geometry: direct and inverse problems. Lecture Notes in Math. vol. 1207 (Springer-Verlag, 1986).Google Scholar
[6]Berger, M., Gauduchon, P., and Mazet, E.Le Spectre d'une Variété Riemannienes. Lecture Notes in Math. vol. 194 (Springer-Verlag 1974).Google Scholar
[7]Bessa, G. P. and Costa, M. S.Eigenvalue estimates for submanifolds with locally bounded mean curvature in N × R. Proc. Amer. Math. Soc. 137 (2009), 1093–1102.CrossRefGoogle Scholar
[8]Bessa, G. P. and Montenegro, J. F.Eigenvalue estimates for submanifolds with locally bounded mean curvature. Ann. Global Anal. Geom. 24 (2003), 279–290.CrossRefGoogle Scholar
[9]Bessa, G. P. and Montenegro, J. F.An extension of Barta's theorem and geometric applications. Ann. Global Anal. Geom. 31 (2007), 345–362.CrossRefGoogle Scholar
[10]Bessa, G. P. and Montenegro, J. F.On Cheng's eigenvalue comparison theorem. Math. Proc. Camb. Phil. Soc. 144 (2008), 673–682.CrossRefGoogle Scholar
[11]Bessa, G. P., Montenegro, J. F., and Piccione, PaoloRiemannian submersions with discrete spectrum. J. Geom. Anal. 22 (2012), 603–620.CrossRefGoogle Scholar
[12]Betz, C., Camera, G. A., and Gzyl, H.Bounds for first eigenvalue of a spherical cap. Appl. Math. Optim. 10 (1983), 193–202.CrossRefGoogle Scholar
[13]Bianchini, B., Mari, L., and Rigoli, M. On some aspects of oscillation theory and geometry to appear in Mem. Amer. Math. Soc.Google Scholar
[14]Candel, A.Eigenvalue estimates for minimal surfaces in Hyperbolic space. Trans. Amer. Math. Soc. 359 (2007), 3567–3575.CrossRefGoogle Scholar
[15]Chavel, I.Eigenvalues in riemannian geometry. Pure Appl. Math. (1984), (Academic Press, inc).Google Scholar
[16]Cheng, S. Y., Li, P., and Yau, S. T.Heat equations on minimal submanifolds and their applications. Amer. J. Math. 106 (1984), 1033–1065.CrossRefGoogle Scholar
[17]Chernoff, P.Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12 (1973), 401–414.CrossRefGoogle Scholar
[18]Cheung, L.-F. and Leung, P.-F.Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space. Math. Z. 236 (2001), 525–530.CrossRefGoogle Scholar
[19]Davies, E. B.Spectral Theory and Differential Operators (Cambridge University Press, 1995).CrossRefGoogle Scholar
[20]Friedland, S. and Hayman, W. K.Eigenvalue inequalities for the dirichlet problem on spheres and the growth of subharmonic functions. Comment. Math. Helv. 51 (1976), 133–161.CrossRefGoogle Scholar
[21]Greene, R. E. and Wu, H.Function theory on manifolds which possess a pole. Lecture Notes in Math. 699 (Springer, Berlin, 1979).Google Scholar
[22]Jorge, L. and Koutrofiotis, D.An estimate for the curvature of bounded submanifolds. Amer. J. Math. 103 (1980), 711–725.CrossRefGoogle Scholar
[23]Li, P.Lecture notes on Geometric analysis. Lecture Notes Series 6 (Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993). iv+90 pp.Google Scholar
[24]Meeks, W. and Rosenberg, H.The theory of minimal surfaces in $M^{2}\times \mathbb{R}$. Comment. Math. Helv. 80 (2005), 811–858.CrossRefGoogle Scholar
[25]Meeks, W. and Rosenberg, H.Stable minimal surfaces in $M^{2}\times \mathbb{R}$. J. Differential Geom. 68 (2004), 515–534.Google Scholar
[26]Persson, A.Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator. Math. Scand. 8 (1960) 143–153.CrossRefGoogle Scholar
[27]Pigola, S., Rigoli, M., and Setti, A. G.Vanishing and finiteness results in geometric analysis. A generalization of the Bochner technique. Progr. Math. 266 (Birkhauser Verlag, Basel, 2008).Google Scholar
[28]Pinsky, M. A.The first eigenvalue of a sphercial cap. Appl. Math. Opt. 7 (1981), 137–139.CrossRefGoogle Scholar
[29]Sato, S.Barta's inequalities and the first eigenvalue of a cap domain of a 2-sphere. Math. Z. 181, (1982), 313–318.CrossRefGoogle Scholar
[30]Strichartz, R. S.Analysis of the Laplacian on the complete Riemannian manifold J. Funct. An. 52 (1983), 48–79.CrossRefGoogle Scholar
[31]Tashiro, Y.Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117 (1965), 251–275.CrossRefGoogle Scholar