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Eigenvalue estimates for submanifolds of warped product spaces

  • G. P. BESSA (a1), S. C. GARCÍA–MARTÍNEZ (a2), L. MARI (a1) and H. F. RAMIREZ–OSPINA (a2)


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 fC(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.



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[1]Barbosa, J. L. and Do Carmo, M. P.Stability of minimal surfaces and eigenvalues of the Laplacian. Math. Z. 173 (1980), 1328.
[2]Barroso, C. S. and Bessa, G. P.A note on the first eigenvalue of spherically symmetric manifolds. Mat. Contemp. 30 (2006), 6369.
[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), 8286.
[4]Barta, J.Sur la vibration fundamentale d'une membrane. C. R. Acad. Sci. 204 (1937), 472473.
[5]Bérard, P. H.Spectral geometry: direct and inverse problems. Lecture Notes in Math. vol. 1207 (Springer-Verlag, 1986).
[6]Berger, M., Gauduchon, P., and Mazet, E.Le Spectre d'une Variété Riemannienes. Lecture Notes in Math. vol. 194 (Springer-Verlag 1974).
[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), 10931102.
[8]Bessa, G. P. and Montenegro, J. F.Eigenvalue estimates for submanifolds with locally bounded mean curvature. Ann. Global Anal. Geom. 24 (2003), 279290.
[9]Bessa, G. P. and Montenegro, J. F.An extension of Barta's theorem and geometric applications. Ann. Global Anal. Geom. 31 (2007), 345362.
[10]Bessa, G. P. and Montenegro, J. F.On Cheng's eigenvalue comparison theorem. Math. Proc. Camb. Phil. Soc. 144 (2008), 673682.
[11]Bessa, G. P., Montenegro, J. F., and Piccione, PaoloRiemannian submersions with discrete spectrum. J. Geom. Anal. 22 (2012), 603620.
[12]Betz, C., Camera, G. A., and Gzyl, H.Bounds for first eigenvalue of a spherical cap. Appl. Math. Optim. 10 (1983), 193202.
[13]Bianchini, B., Mari, L., and Rigoli, M. On some aspects of oscillation theory and geometry to appear in Mem. Amer. Math. Soc.
[14]Candel, A.Eigenvalue estimates for minimal surfaces in Hyperbolic space. Trans. Amer. Math. Soc. 359 (2007), 35673575.
[15]Chavel, I.Eigenvalues in riemannian geometry. Pure Appl. Math. (1984), (Academic Press, inc).
[16]Cheng, S. Y., Li, P., and Yau, S. T.Heat equations on minimal submanifolds and their applications. Amer. J. Math. 106 (1984), 10331065.
[17]Chernoff, P.Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12 (1973), 401414.
[18]Cheung, L.-F. and Leung, P.-F.Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space. Math. Z. 236 (2001), 525530.
[19]Davies, E. B.Spectral Theory and Differential Operators (Cambridge University Press, 1995).
[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), 133161.
[21]Greene, R. E. and Wu, H.Function theory on manifolds which possess a pole. Lecture Notes in Math. 699 (Springer, Berlin, 1979).
[22]Jorge, L. and Koutrofiotis, D.An estimate for the curvature of bounded submanifolds. Amer. J. Math. 103 (1980), 711725.
[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.
[24]Meeks, W. and Rosenberg, H.The theory of minimal surfaces in $M^{2}\times \mathbb{R}$. Comment. Math. Helv. 80 (2005), 811858.
[25]Meeks, W. and Rosenberg, H.Stable minimal surfaces in $M^{2}\times \mathbb{R}$. J. Differential Geom. 68 (2004), 515534.
[26]Persson, A.Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator. Math. Scand. 8 (1960) 143153.
[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).
[28]Pinsky, M. A.The first eigenvalue of a sphercial cap. Appl. Math. Opt. 7 (1981), 137139.
[29]Sato, S.Barta's inequalities and the first eigenvalue of a cap domain of a 2-sphere. Math. Z. 181, (1982), 313318.
[30]Strichartz, R. S.Analysis of the Laplacian on the complete Riemannian manifold J. Funct. An. 52 (1983), 4879.
[31]Tashiro, Y.Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117 (1965), 251275.

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Eigenvalue estimates for submanifolds of warped product spaces

  • G. P. BESSA (a1), S. C. GARCÍA–MARTÍNEZ (a2), L. MARI (a1) and H. F. RAMIREZ–OSPINA (a2)


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