Eigenvalue estimates for submanifolds with locally bounded mean curvature in 𝑁×ℝ

Bessa, G.; Costa, M.

doi:10.1090/S0002-9939-08-09680-9

Eigenvalue estimates for submanifolds with locally bounded mean curvature in $N \times \mathbb {R}$
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by G. Pacelli Bessa and M. Silvana Costa

Proc. Amer. Math. Soc. 137 (2009), 1093-1102

DOI: https://doi.org/10.1090/S0002-9939-08-09680-9

Published electronically: October 21, 2008

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Abstract:

We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $N \times \mathbb {R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N} \leq \kappa$. When the immersion is minimal our estimates are sharp. We also show that cylindrically bounded minimal surfaces have a positive fundamental tone.

References

J. Barta, Sur la vibration fundamentale d’une membrane. C. R. Acad. Sci. 204 (1937), 472–473.
Pierre H. Bérard, Spectral geometry: direct and inverse problems, Lecture Notes in Mathematics, vol. 1207, Springer-Verlag, Berlin, 1986. With appendixes by Gérard Besson, and by Bérard and Marcel Berger. MR 861271, DOI 10.1007/BFb0076330
Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
G. Pacelli Bessa and J. Fábio Montenegro, Eigenvalue estimates for submanifolds with locally bounded mean curvature, Ann. Global Anal. Geom. 24 (2003), no. 3, 279–290. MR 1996771, DOI 10.1023/A:1024750713006
G. Pacelli Bessa and J. Fábio Montenegro, An extension of Barta’s theorem and geometric applications, Ann. Global Anal. Geom. 31 (2007), no. 4, 345–362. MR 2325220, DOI 10.1007/s10455-007-9058-8
G. Pacelli Bessa and J. Fabio Montenegro, On compact $H$-hypersurfaces of $N\times \Bbb R$, Geom. Dedicata 127 (2007), 1–5. MR 2338510, DOI 10.1007/s10711-007-9145-9
Alberto Candel, Eigenvalue estimates for minimal surfaces in hyperbolic space, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3567–3575. MR 2302506, DOI 10.1090/S0002-9947-07-04104-9
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Shiu Yuen Cheng, Peter Li, and Shing-Tung Yau, Heat equations on minimal submanifolds and their applications, Amer. J. Math. 106 (1984), no. 5, 1033–1065. MR 761578, DOI 10.2307/2374272
Leung-Fu Cheung and Pui-Fai Leung, Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space, Math. Z. 236 (2001), no. 3, 525–530. MR 1821303, DOI 10.1007/PL00004840
Thomas Hasanis, Isometric immersions into spheres, J. Math. Soc. Japan 33 (1981), no. 3, 551–555. MR 620291, DOI 10.2969/jmsj/03330551
Luquésio P. de M. Jorge and Frederico Xavier, A complete minimal surface in $\textbf {R}^{3}$ between two parallel planes, Ann. of Math. (2) 112 (1980), no. 1, 203–206. MR 584079, DOI 10.2307/1971325
F. J. López, F. Martín, and S. Morales, Adding handles to Nadirashvili’s surfaces, J. Differential Geom. 60 (2002), no. 1, 155–175. MR 1924594
F. J. López, Francisco Martin, and Santiago Morales, Complete nonorientable minimal surfaces in a ball of $\Bbb R^3$, Trans. Amer. Math. Soc. 358 (2006), no. 9, 3807–3820. MR 2219000, DOI 10.1090/S0002-9947-06-04004-9
Francisco Martín and Santiago Morales, A complete bounded minimal cylinder in $\Bbb R^3$, Michigan Math. J. 47 (2000), no. 3, 499–514. MR 1813541, DOI 10.1307/mmj/1030132591
William H. Meeks and Harold Rosenberg, The theory of minimal surfaces in $M\times \Bbb R$, Comment. Math. Helv. 80 (2005), no. 4, 811–858. MR 2182702, DOI 10.4171/CMH/36
William H. Meeks III and Harold Rosenberg, Stable minimal surfaces in $M\times \Bbb R$, J. Differential Geom. 68 (2004), no. 3, 515–534. MR 2144539
Nikolai Nadirashvili, Hadamard’s and Calabi-Yau’s conjectures on negatively curved and minimal surfaces, Invent. Math. 126 (1996), no. 3, 457–465. MR 1419004, DOI 10.1007/s002220050106
R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S. Y. Cheng; With a preface translated from the Chinese by Kaising Tso. MR 1333601