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 [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 \(n^{2}\).

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.