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Fundamental tone of minimal hypersurfaces with finite index in hyperbolic space
- Keomkyo Seo^{1}Email author
https://doi.org/10.1186/s13660-016-1071-7
© Seo 2016
- Received: 22 January 2016
- Accepted: 19 April 2016
- Published: 27 April 2016
Abstract
Let M be a complete minimal hypersurface in hyperbolic space \(\mathbb {H}^{n+1}(-1)\) with constant sectional curvature −1. We prove that if M has a finite index and finite \(L^{2}\) norm of the second fundamental form, then the fundamental tone \(\lambda_{1} (M)\) is bounded above by \(n^{2}\).
Keywords
- minimal hypersurface
- finite index
- hyperbolic space
- fundamental tone
- eigenvalue
MSC
- 53C40
- 53C42
1 Introduction
Theorem
[2]
There have been extensive investigations to obtain an upper bound for the fundamental tone of complete minimal submanifolds in hyperbolic space. Castillon [5] proved that the spectrum of the Laplacian on a complete minimal hypersurface with finite \(L^{n}\) norm of the second fundamental form in \(\mathbb{H}^{n+1}\), denoted by \(\operatorname {Spec}(\Delta)\), is given by \(\operatorname {Spec}(\Delta) = [\frac{(n-1)^{2}}{4}, +\infty )\). Candel [6] was able to prove that the fundamental tone of complete simply connected stable minimal surfaces in \(\mathbb{H}^{3} (-1)\) is at most \(\frac{4}{3}\). In [7], the author proved that if M is a complete stable minimal hypersurface in \(\mathbb {H}^{n+1}(-1)\) with finite \(L^{2}\) norm of the second fundamental form, then \(\frac{(n-1)^{2}}{4} \leq\lambda_{1} (M) \leq n^{2}\). Later, Bérard et al. [8] improved the upper bound for complete stable minimal surfaces in \(\mathbb{H}^{3}(-1)\). Indeed, they proved that the fundamental tone of complete stable minimal surfaces in \(\mathbb{H}^{3} (-1)\) is at most \(\frac{4}{7}\). Fu and Tao [9] showed that if M is an n-dimensional complete submanifold in \(\mathbb{H}^{m}(-1)\) with parallel mean curvature vector H and with finite \(L^{p}\) norm of the traceless second fundamental form for \(p \geq n\), then \(\lambda_{1} (M)\) is less than or equal to \(\frac {(n-1)^{2}(1-\vert H\vert ^{2})}{4}\). Recently, Gimeno [10] proved that if \(M^{2}\) is a complete minimal surface in \(\mathbb{H}^{m} (-1)\) with finite \(L^{2}\) norm of the second fundamental form, then \(\lambda_{1} (M)=\frac{1}{4}\).
The aim of this paper is to obtain an upper bound for the fundamental tone of complete minimal hypersurfaces in \(\mathbb{H}^{n+1}(-1)\) with finite index and finite \(L^{2}\) norm of the second fundamental form. More precisely, we prove the following.
Theorem 1.1
It is obvious that a complete stable minimal hypersurface in \(\mathbb{H}^{n+1}(-1)\) has index 0. Hence our theorem can be regarded as an extension of the results in [6–8]. When \(n=2\), we remark that the finite index condition can be omitted, since the finiteness of the \(L^{2}\) norm of the second fundamental form implies that M has finite index, which was proved by Bérard et al. [11]. However, in this case, our theorem is weaker than Theorem 4.1 in [5] or Theorem A in [10].
2 Proof of Theorem 1.1
In this section, we prove our main theorem.
Proof of Theorem 1.1
The lower bound of \(\lambda_{1} (M)\) is given by \(\frac{(n-1)^{2}}{4}\), which was done by Cheung and Leung [2] as mentioned in the Introduction. Thus it suffices to prove that the upper bound of \(\lambda_{1}(M)\) is \(n^{2}\).
Declarations
Acknowledgements
The author would like to thank the referees for careful reading of the manuscript and many helpful suggestions. This research was supported in part by the Sookmyung Women’s University Research Grants (1-1403-0097).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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