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Fundamental tone of minimal hypersurfaces with finite index in hyperbolic space

Journal of Inequalities and Applications20162016:127

https://doi.org/10.1186/s13660-016-1071-7

  • Received: 22 January 2016
  • Accepted: 19 April 2016
  • Published:

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

McKean [1] proved that the fundamental tone of an n-dimensional complete simply connected Riemannian manifold M with sectional curvature bounded above by \(-\kappa^{2}<0\) is bigger than or equal to \(\frac{(n-1)^{2}\kappa^{2}}{4}\), where κ is a real number. Moreover, his result is sharp since the equality is attained by the hyperbolic space \(\mathbb{H}^{n}(-\kappa^{2})\) with constant sectional curvature \(-\kappa^{2}\). We recall that the fundamental tone \(\lambda_{1}(M)\) is defined by
$$\begin{aligned} \lambda_{1} (M) = \inf \biggl\{ \frac{\int_{M} \vert \nabla f\vert ^{2}}{\int_{M} f^{2}}: 0\neq f \in W^{1,2}_{0} (M) \biggr\} . \end{aligned}$$
Interestingly, Cheung and Leung [2] obtained the same lower bound for the fundamental tone of complete submanifold in \(\mathbb {H}^{m}(-\kappa^{2})\) with bounded mean curvature as follows (see also [3, 4]).

Theorem

[2]

Let M be an n-dimensional complete noncompact submanifold in \(\mathbb{H}^{m}(-\kappa^{2})\) with the mean curvature vector H. If \(\vert H\vert \leq\alpha< n-1\), then
$$\begin{aligned} \lambda_{1}(M) \geq\frac{(n-1-\alpha)^{2} \kappa^{2}}{4}. \end{aligned}$$

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

Let M be a complete orientable minimal hypersurface in \(\mathbb {H}^{n+1}(-1)\) with \(\int_{M} \vert A\vert ^{2} < \infty\). Suppose M has finite index. Then we have
$$\begin{aligned} \frac{(n-1)^{2}}{4} \leq \lambda_{1}(M) \leq n^{2}. \end{aligned}$$

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 [68]. 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 \(n^{2}\).

Since M has a finite index, there exists a compact subset \(K \subset M\) such that \(M \setminus K\) is stable (see [12] for example), i.e., for any compactly supported Lipschitz function f on \(M\setminus K\),
$$\begin{aligned} \int_{M \setminus K} \vert \nabla f\vert ^{2} - \bigl( \vert A\vert ^{2}-n\bigr)f^{2}\,dv \geq0, \end{aligned}$$
(1)
where \(\vert A\vert ^{2}\) denotes the squared length of the second fundamental form on M and dv denotes the volume form for the induced metric on M. Note that, for some geodesic ball \(B(R_{0}) \subset M\) centered at \(p\in M\) of radius \(R_{0}\) containing the compact set K, the region \(M \setminus B(R_{0})\) is still stable. Thus, without loss of generality, we may assume that \(K=B(R_{0})\).
Choose a geodesic ball \(B(R)\subset M\) centered at \(p\in M\) of radius \(R>R_{0}\) and take a cut-off function \(0\leq\phi\leq1\) on M satisfying
$$ \phi= \textstyle\begin{cases} 0& \text{on }B(R_{0}),\\ 1& \text{on }B(2R+R_{0})\setminus B(R+R_{0}),\\ 0& \text{on }M \setminus B(3R+R_{0}), \end{cases} $$
and \(\vert \nabla\phi \vert \leq\frac{1}{R}\) on M. By the definition of the fundamental tone and the domain monotonicity of the eigenvalue, we see that
$$\begin{aligned} \lambda_{1} (M) \leq\lambda_{1} \bigl(M\setminus B(R_{0})\bigr) \leq\frac{\int _{M\setminus B(R_{0})} \vert \nabla f\vert ^{2}}{\int_{M\setminus B(R_{0})} f^{2}} \end{aligned}$$
for any \(f \in W^{1,2}_{0} (M \setminus B(R_{0}))\). Substituting f with \(\vert A\vert \phi\) gives
$$\begin{aligned} &\lambda_{1}(M) \int_{M\setminus B(R_{0})} \vert A\vert ^{2} \phi^{2} \\ &\quad \leq \int_{M\setminus B(R_{0})} \bigl\vert \nabla\bigl(\vert A\vert \phi\bigr) \bigr\vert ^{2} \\ &\quad = \int_{M\setminus B(R_{0})} \phi^{2} \bigl\vert \nabla \vert A \vert \bigr\vert ^{2} + \int_{M\setminus B(R_{0})} \vert A\vert ^{2}\vert \nabla\phi \vert ^{2} +2 \int_{M\setminus B(R_{0})} \vert A\vert \phi \bigl\langle \nabla \vert A \vert , \nabla\phi\bigr\rangle . \end{aligned}$$
Using the Schwarz inequality and the geometric-arithmetic mean inequality, we get
$$\begin{aligned} 2 \int_{M\setminus B(R_{0})} \vert A\vert \phi \bigl\langle \nabla \vert A \vert , \nabla\phi \bigr\rangle \leq\varepsilon \int_{M\setminus B(R_{0})} \vert A\vert ^{2}\vert \nabla\phi \vert ^{2} + \frac{1}{\varepsilon} \int_{M\setminus B(R_{0})} \phi^{2}\bigl\vert \nabla \vert A\vert \bigr\vert ^{2} \end{aligned}$$
for any \(\varepsilon>0\). Therefore
$$\begin{aligned} \lambda_{1}(M) \int_{M\setminus B(R_{0})} \vert A\vert ^{2} \phi^{2} \leq& (1+\varepsilon) \int_{M\setminus B(R_{0})} \vert A\vert ^{2}\vert \nabla\phi \vert ^{2} \\ &{}+ \biggl(1+\frac{1}{\varepsilon} \biggr) \int_{M\setminus B(R_{0})} \phi^{2}\bigl\vert \nabla \vert A\vert \bigr\vert ^{2}. \end{aligned}$$
(2)
On the other hand, a Simons-type inequality [13, 14] for minimal hypersurfaces in \(\mathbb{H}^{n+1}\) asserts that
$$\begin{aligned} \vert A\vert \Delta \vert A\vert + \vert A\vert ^{4} + n \vert A\vert ^{2} = \vert \nabla A\vert ^{2} - \bigl\vert \nabla \vert A\vert \bigr\vert ^{2}. \end{aligned}$$
Applying the Kato inequality [15],
$$\begin{aligned} \vert \nabla A\vert ^{2} - \bigl\vert \nabla \vert A\vert \bigr\vert ^{2} \geq\frac{2}{n} \bigl\vert \nabla \vert A \vert \bigr\vert ^{2}, \end{aligned}$$
we have
$$\begin{aligned} \vert A\vert \Delta \vert A\vert + \vert A\vert ^{4} + n \vert A\vert ^{2} \geq\frac{2}{n} \bigl\vert \nabla \vert A\vert \bigr\vert ^{2}. \end{aligned}$$
Multiplying both sides by the function \(\phi^{2}\) and integrating over \(B(3R+R_{0})\setminus B(R_{0})\), we get
$$\begin{aligned} \frac{2}{n} \int_{M\setminus B(R_{0})} \phi^{2} \bigl\vert \nabla \vert A \vert \bigr\vert ^{2} \leq& \int_{M\setminus B(R_{0})} \phi^{2} \vert A\vert ^{4} + n \int_{M\setminus B(R_{0})} \phi^{2} \vert A\vert ^{2} \\ &{}- \int_{M\setminus B(R_{0})} \phi^{2} \bigl\vert \nabla \vert A \vert \bigr\vert ^{2} -2 \int_{M\setminus B(R_{0})} \vert A\vert \phi\bigl\langle \nabla \vert A \vert , \nabla\phi\bigr\rangle , \end{aligned}$$
(3)
where we used the divergence theorem.
Replacing f with \(\phi \vert A\vert \) in the stability inequality (1) on \(M \setminus B(R_{0})\) gives
$$\begin{aligned} \int_{M\setminus B(R_{0})} \bigl\vert \nabla\bigl(\phi \vert A\vert \bigr) \bigr\vert ^{2} \geq \int_{M\setminus B(R_{0})} \bigl(\vert A\vert ^{2} -n\bigr)\vert A\vert ^{2}\phi^{2}, \end{aligned}$$
which implies
$$\begin{aligned} &\int_{M\setminus B(R_{0})} \vert A\vert ^{2}\vert \nabla\phi \vert ^{2} + \int_{M\setminus B(R_{0})} \phi^{2}\bigl\vert \nabla \vert A\vert \bigr\vert ^{2} + 2 \int_{M\setminus B(R_{0})} \vert A\vert \phi\bigl\langle \nabla \vert A \vert , \nabla\phi\bigr\rangle \\ &\quad \geq \int_{M\setminus B(R_{0})} \vert A\vert ^{4} \phi^{2} - n \int_{M\setminus B(R_{0})} \vert A\vert ^{2} \phi^{2} . \end{aligned}$$
(4)
Combining (3) with (4), we obtain
$$\begin{aligned} \frac{2}{n} \int_{M\setminus B(R_{0})} \phi^{2}\bigl\vert \nabla \vert A\vert \bigr\vert ^{2} \leq \int _{M\setminus B(R_{0})} \vert A\vert ^{2}\vert \nabla\phi \vert ^{2} + 2n \int_{M\setminus B(R_{0})} \vert A\vert ^{2}\phi^{2}. \end{aligned}$$
(5)
Hence, using (2) and (5), we have
$$\begin{aligned} 2 \biggl\{ \frac{1}{n} - \frac{n(1+\frac{1}{\varepsilon})}{\lambda_{1}(M)} \biggr\} \int _{M\setminus B(R_{0})} \phi^{2}\bigl\vert \nabla \vert A \vert \bigr\vert ^{2} \leq \biggl\{ 1 + \frac {2n(1+\varepsilon)}{\lambda_{1}(M)} \biggr\} \int_{M\setminus B(R_{0})} \vert A\vert ^{2}\vert \nabla\phi \vert ^{2}. \end{aligned}$$
(6)
We now suppose that \(\lambda_{1} (M) >n^{2}\). For a sufficiently large \(\varepsilon>0\), letting \(R\rightarrow\infty\) in (6) shows that \(\vert \nabla \vert A\vert \vert \equiv0\) on \(M\setminus B(R_{0})\), which implies that \(\vert A\vert \) is constant on \(M\setminus B(R_{0})\). Since the volume of any complete minimal hypersurface in hyperbolic space is infinite and \(L^{2}\) norm of \(\vert A\vert \) is finite by our assumption, we see that \(\vert A\vert \equiv0\) outside the compact subset \(B(R_{0})\). It follows from the maximum principle for minimal hypersurfaces in \(\mathbb{H}^{n+1}\) that M must be totally geodesic. However, due to McKean [1], the fundamental tone of totally geodesic hyperplanes in \(\mathbb{H}^{n+1}\) is equal to \(\frac{(n-1)^{2}}{4}\), which gives a contradiction. Therefore we get the conclusion. □

Remark 2.1

The proof of Theorem 1.1 relies on the inequality (6), which is called a Caccioppoli-type inequality. In [16], Ilias et al. intensively studied a Caccioppoli-type inequality on constant mean curvature hypersurfaces in Riemannian manifolds.

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

(1)
Department of Mathematics, Sookmyung Women’s University, Cheongpa-ro 47-gil 100, Yongsan-ku, Seoul, 04310, Korea

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