The number of cusps of complete Riemannian manifolds with finite volume

TL;DR

This paper establishes bounds on the number of cusps in complete Riemannian manifolds with finite volume, using volume decay estimates, comparison theorems, and nonlinear $p$-Laplacian theory.

Contribution

It provides new upper bounds on cusp counts based on volume and geometric conditions, extending previous results to measure spaces and nonlinear analysis.

Findings

Number of cusps is bounded by volume under certain conditions

Volume decay estimates are crucial for cusp counting

Upper bounds are derived using $p$-Laplacian theory

Abstract

In this paper, we will count the number of cusps of complete Riemannian manifolds $M$ with finite volume. When $M$ is a complete smooth metric measure spaces, we show that the number of cusps in bounded by the volume $V$ of $M$ if some geometric conditions hold true. Moreover, we use the nonlinear theory of the $p$-Laplacian to give a upper bound of the number of cusps on complete Riemannian manifolds. The main ingredients in our proof are a decay estimate of volume of cusps and volume comparison theorems.

Equations

$${\lambda }_1(M)\le \frac{(n-1{)}^2}{4}.$$

$${\mu }_1(M)=\underset{\varphi \in H^1(M)),{\int }_M\varphi =0}{\inf }\frac{{\int }_M|\nabla \varphi {|}^2}{\int {\varphi }^2}.$$

$${\mu }_1(M)\le \max \{{\lambda }_1({\Omega }_1),{\lambda }_1({\Omega }_2)\}$$

$${\mu }_1(M)\ge \frac{(n-1{)}^2}{4}.$$

$$N(M)\leq C(n)\Big{(}\frac{V}{V_{o}(1)}\Big{)}^{2}\ln\frac{V}{V_{o}(1)}$$

$${\Delta }_f\cdot =\Delta \cdot +⟨\nabla f,\nabla \cdot ⟩$$

$$Ri{c}_f:=Ri{c}_M+Hessf$$

$$|f|(x)\le \alpha r(x)+\beta$$

$$Ri{c}_f\ge -(n-1)$$

$${\mu }_1(M)\ge \frac{(n-1+\alpha {)}^2}{4}$$

$$N(M)\leq C(n)\Big{(}\dfrac{V_{f}}{V_{o,f}(1)}\Big{)}^{2}\ln\Big{(}\dfrac{V_{f}}{V_{o,f}(1)}\Big{)}$$

$${\Delta }_{p}u=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u).$$

$${\Delta }_{p}u=-\lambda |u{|}^{p-2}u$$

$${\mu }_{1,p}(M)=\inf \left\{\frac{{\int }_M|\nabla \varphi {|}^p}{\int {\varphi }^p};\varphi \in W^{1,p}(M),\varphi \ne 0\text{ and }{\int }_M|\varphi {|}^{p-1}\varphi =0\right\}.$$

$${\mu }_{1,p}(M)\le \max \{{\lambda }_{1,p}({\Omega }_1),{\lambda }_{1,p}({\Omega }_2)\}$$

$$\lambda_{1,p}(M)\leq\big{(}\frac{n-1}{p}\Big{)}^{p}.$$

$$Ri{c}_M\ge -(n-1)$$

$${\mu }_{1,p}(M)\ge \frac{(n-1{)}^p}{p^p}.$$

$$N(M)\leq C(n,p)\Big{(}\frac{V}{V_{o}(1)}\Big{)}^{2}\ln^{\alpha(p)}\Big{(}\frac{V}{V_{o}(1)}\Big{)}$$

$$d{s_{\rho }}^2=\rho d{s}^2.$$

$$r_{\rho }(x,y)=\underset{\gamma }{\inf }{\ell }_{\rho }(\gamma ),$$

$$B_{\rho }(o,R)=\{x\in M|r_{\rho }(o,x)<R\}$$

$$B(o,R)=\{x\in M|r(o,x)<R\}$$

$${\int }_E\rho {\varphi }^2{e}^{-f}\le {\int }_E|\nabla \varphi {|}^2{e}^{-f}-{\int }_E\mu {\varphi }^2{e}^{-f}$$

$${\Delta }_{f}u\ge -\mu u$$

$${\int }_{E_{\rho }(R)}\rho u^2{e}^{-2r_{\rho }}{e}^{-f}=o(R)$$

$${\int }_{E_{\rho (R+1)}\setminus E_{\rho (R)}}\rho u^2{e}^{-f}\le C(1+(R-R_0{)}^{-1})e^{-2R}{\int }_{E_{\rho }(R_0+1)\setminus E_{\rho }(R_0)}{e}^{2r_{\rho }}{u}^2{e}^{-f}$$

$${\int }_E|\nabla (\varphi e^{2h}u){|}^2{e}^{-f}$$

$$=$$

Full text

1. Introduction

Let $E$ be an end of a Riemannian manifold $M^{n}$ and $\lambda_{1}(M)$ be the first Dirichlet eigenvalue of the Laplacian on $M$. It is well-known that information of $\lambda_{1}(M)$ tells us some geometric properties of the manifold. For example, if $\lambda_{1}(M)>0$ then $M$ must have infinite volume, or if $\lambda_{1}(E)>0$ then either $E$ has finite volume, namely $E$ is a cusp; or $E$ is non-parabolic end with volume of exponent growth. In [Cheng75], Cheng considered complete manifolds $(M^{n},g)$ of dimension $n$ with $Ric_{M}\geq-(n-1)$ and gave an upper bound of $\lambda_{1}(M)$

[TABLE]

Later, Li and Wang showed in [LW02] that if $Ric_{M}\geq-(n-1)$ and $\lambda_{1}(M)$ is maximal then either $M$ has only one end or; $M$ is a topological cylinder with certain warped metric product. Since $\lambda_{1}(M)$ is maximal, $M$ must have infinite volume. Hence, Li-Wang’s result says that we can count the number of ends of complete Riemannian manifold $M$ with $Ric_{M}\geq-(n-1)$ provided $\lambda_{1}(M)$ obtains its maximal value. In this case,$M$ has at most two ends.

Interestingly, when $M$ is a complete Riemannian manifold of finite volume, Li and Wang proved in [LW10] that one can count ends (cusps) via the bottom of Neumann spectrum $\mu_{1}(M)$ defined by

[TABLE]

Note that $\mu_{1}(M)$ plays the role of a generalized first non-zero Neumann eigenvalue, although $\mu_{1}(M)$ might not necessarily be an eigenvalue. However, if $M$ is compact then $\mu_{1}(M)$ is itself a positive eigenvalue called the first (Neumann) eigenvalue of the Laplacian (see [Veron91] or also see [Matei00]). As in [Veron91, LW10], variational principle implies

[TABLE]

for any two disjoint domains $\Omega_{1}$ and $\Omega_{2}$ of $M$, where $\lambda_{1}(\Omega_{1})$ and $\lambda_{1}(\Omega_{2})$ are their first Dirichlet eigenvalues respectively. Li Wang counted cusps on complete manifolds with finite volume as follows.

Theorem 1.1** ([LW10]).**

Let $M^{n}$ be a complete Riemannian manifold with Ricci curvature bounded from below by $Ric_{M}\geq-(n-1)$ Assume that $M$ has finite volume given by $V$, and

[TABLE]

Let us denote $N(M)$ to be the number of ends (cusps) of M. Then there exists a constant $C(n)>0$ depending only on $n$, such that,

[TABLE]

where $V_{o}(1)$ denotes the volume of the unit ball centered at any point $o\in M$.

Due to Cheng’s upper bound of $\lambda_{1}(M)$ and variational characteristic of $\mu_{1}(M)$, one can see that in Theorem 1.1, $\mu_{1}(M)$ is maximal. This means that instead of $\lambda_{1}(M)$, we can use $\mu_{1}(M)$ to count the number of cusps of complete Riemannian manifolds with finite volume. We would like to mention that to count cusps of complete manifolds, Li and Wang used a decay volume estimate and a volume comparison theorem.

Motivated by the beauty of Theorem 1.1, in this paper, we want to estimate the number of cusps on smooth metric measure spaces $(M^{n},g,e^{-f}dv)$ with finite $f$-volume $V_{f}$. Recall that a smooth metric measure space $(M^{n},g,e^{-f}dv)$ is a complete Riemannian manifold $(M^{n},g)$ of dimension $n$ with $f\in\mathcal{C}^{\infty}(M)$ is a smooth weighted function and $e^{-f}dv$ is the weighted volume. Here $dv$ is the volume form with respect to the metric $g$. On $(M,g,e^{-f}dv$, we consider the weighted Laplacian

[TABLE]

which is a self-adjoint operator. Associated to the weighted Laplacian, we define the Bakry-Émery curvature by

[TABLE]

where $Ric_{M}$ is the Ricci curvature of $M$ and $Hessf$ is the Hessian of $f$. Following the same strategy as in [LW10], we first give a decay estimate for weighted volume and use a volume comparison theorem in [MW12] to prove the below theorem.

Theorem 1.2**.**

Let $(M,g,e^{f}dv)$ be a complete smooth metric measure space. Assume for some nonnegative constants $\alpha$ and $\beta$,

[TABLE]

for $x\in M$ with Ricci curvature bounded from below by

[TABLE]

Assume that $M$ has finite volume given by $V_{f}$, and

[TABLE]

Let us denote $N(M)$ to be the number of ends (cusps) of $M$. Then there exists a contant $C(n)>0$ depending only on $n$, such that,

[TABLE]

where $V_{o,f}(1)$ denotes the $f$-volume of the unit ball centered at any fixed point $o\in M$.

On the other hand, our second aim in this paper is to count the number of cusps on complete Riemannian manifold $(M^{n},g)$ with finite volume via the nonlinear theory of $p$-Laplacian, for $1\leq p<\infty$. On a complete Riemannian manifold, for any $u\in W^{1,p}_{loc}$, the $p$-Laplacian denoted by $\Delta_{p}$ acting on $u$ as follows

[TABLE]

If $\lambda$ satisfies

[TABLE]

then $\lambda$ is said to be an eigenvalue of $p$-Laplacian and $u$ is called an eigenfunction with respect to $\lambda$. As in [Matei00] and [Veron91], we define

[TABLE]

For any couple of nonempty disjoint open subsets $\Omega_{1},\Omega_{2}$ of $M$, Veron proved, in his paper [Veron91], (see also [Matei00]) the following result

[TABLE]

where $\lambda_{1,p}(\Omega_{1})$ and $\lambda_{1,p}(\Omega_{2})$ are their first Dirichlet $p$-eigenvalues, respectively. As in [Dung17, SW14] we know that if $M$ satisfies $Ric_{M}\geq-(n-1)$ then $\lambda_{1,p}$ is bounded from upper by

[TABLE]

For further discussion on $p$-Laplacian and its eigenvalues, we refer the reader to [Dung17, Matei00, SW14, Veron91] and the references therein. Now, we can count the number of cusps as follows.

Theorem 1.3**.**

Let $(M^{n},g)$ be a complete with finite volume given by $V$. Assume that the Ricci curvature is bounded from below by

[TABLE]

and

[TABLE]

Let us denote $N(M)$ to be the number of ends (cusps) of $M$. Then there exists a contant $C(n,p)>0$ depending only on $n$ and $p$ such that,

[TABLE]

*where $\alpha(p)=\max\{1,p/2\}$ and $V_{o}(1)$ denotes the volume of the unit ball centered at a fixed point $o\in M$. *

As noticed in the previous part, two main ingredients in our proof are a decay estimate for volume and a comparison volume theorem. Moreover, it is also worth to note that $\mu_{1,p}(M)$ in Theorem 1.3 is maximal. When $\lambda_{1,p}(M)$ is maximal, it is proved in [Dung17, SW14] that $M$ has at most two ends. In fact, in [Dung17, SW14], the authors pointed out that either $M$ must have only one end or; $M$ is a topological cylinder provided that $Ric_{M}\geq-(n-1)$ and $\lambda_{1,p}$ is maximal.

The paper is organized as follows. In the Section 2, we introduce a estimate of volume decay rate on smooth metric measure spaces which can be considered as a generalization of volume decay rate in [LW06, LW10]. In the Section 3, we will count the number of cusps on complete smooth metric measure spaces with finite volume. Finally, we use the nonlinear theory of $p$-Laplacian in Section 4 to estimate the number of cusps on Riemannian manifolds. Several complete manifolds of finite volume are investigated in this section.

2. Smooth metric measure spaces with weighted Poincaré inequality

Let $(M^{n},g,e^{-f}dv)$ be a smooth metric measure space with a weighted positive function $\rho\in\mathcal{C}(M)$. We define the $\rho$-metric by

[TABLE]

Thanks to this metric, we can define the $\rho-$distance function to be

[TABLE]

where the infimum is taken over all smooth curves $\gamma$ joining $x$ and $y$, and $\ell$ is the length of $\gamma$ with respect to $ds^{2}_{\rho}$. For a fixed point $o\in M$, we denote $r_{\rho}(x)=r_{\rho}(o,x)$ to be the $\rho-$distance to $o$. As in [LW06], we know that $|\nabla r_{\rho}|^{2}(x)=\rho(x)$.

Throughout this article, we denote

[TABLE]

to be the geodesic ball centered at $o\in M$ with radius $R$. We also denote the geodesic ball

[TABLE]

to be the set of points in $M$ that has distance less than $R$ from point $o$ with respect to the background metric $ds^{2}_{M}$. For simple the notation, sometime, we will suppress the dependency of $o$ and write $B_{\rho}(R)=B_{\rho}(o,R)$ and $B(R)=B(o,R)$. Finally, suppose that $E$ is an end of $M$, we denote $E_{\rho}(R)=B_{\rho}(R)\cap E$.

Lemma 2.1**.**

Let $(M,ds^{2}_{\rho},e^{-f}dv)$ be a complete smooth metric measure space. Suppose $E$ is an end of $M$ satisfying there exists a nonnegative function $\rho(x)$ defined on $E$ with the property that

[TABLE]

for any compactly supported function $\phi\in C_{c}^{\infty}(E)$ and $\mu$ be a function defined on $E$. Let $u$ be a nonnegative function defined on $E$ such that the differential inequality

[TABLE]

holds true. If $u$ has the growth condition

[TABLE]

as $R\to\infty$, then it must satisfy the decay estimate

[TABLE]

for some constant $C>0$.

Proof of Lemma 2.1.

To prove the Lemma 2.1, we will combine both arguments in [LW06] and [LW10]. Let $\phi(r_{\rho}(x))$ be a nonnegative cut-off function with support in $E$, where $r_{\rho}(x)$ is the $\rho-$distance to the fixed point $p$. Then for any function $h(r_{\rho}(x))$, integration by parts implies

[TABLE]

On the other hand, by assumption, we have

[TABLE]

hence using (2.1), we obtain

[TABLE]

From now, we will devided the proof into three step.

Step 1: We claim that for any $0<\delta<1$, there exists a constant $0<C<\infty$ such that,

[TABLE]

Indeed, let us now choose

[TABLE]

It is easy to see that

[TABLE]

Moreover, we also choose

[TABLE]

for some fixed $K>(R_{0}+1)(1+\delta)$. When $R\geq\frac{K}{1+\delta}$, we have

[TABLE]

and

[TABLE]

Substituting into (2.2), we infer

[TABLE]

Therefore, by rearrangement the above inequality, we obtain

[TABLE]

Thus

[TABLE]

Due to the definition of $h$ and the assumption on the growth condition of $u$, we see that the last two terms on the right hand side tend to as $R\to\infty$. Hence, we have the estimate

[TABLE]

Note that the right hand side does not depend on $K$, by letting $K\to\infty$ we conclude that

[TABLE]

where

[TABLE]

**Step 2: **We want to prove that

[TABLE]

To do this, our first aim is to improve (2.3) by setting $h=r_{\rho}$ in the previous arguments. Now, suppose that $h=r_{\rho}$, by (2.2), we infer

[TABLE]

For $R_{0}<R_{1}<R$, we choose

[TABLE]

Plugging $\phi$ in the above inequality, we obtain

[TABLE]

Observe that for any $0<t<R-R_{1},$ the following inequality

[TABLE]

holds true. Therefore, we conclude that

[TABLE]

Now, by taking $R_{1}=R_{0}+1,t=1,$ and setting

[TABLE]

the inequality (2.5) becomes

[TABLE]

where

[TABLE]

is independent of $R$. Iterating this inequality, we obtain that for any positive integer $k$ and $R\geq 1$

[TABLE]

for some constant $C_{2}$. Note that in our previous estimate (2.3), we have proved the following inequality

[TABLE]

for any $\delta<1$. Thus, this implies that

[TABLE]

Now, if we choose $2(1-\delta)<\ln 2$ then

[TABLE]

as $k\to\infty$. Consequently, we obtain

[TABLE]

Therefore, by adjusting the constant, we conclude

[TABLE]

for all $R\geq R_{0}$.

Finally, using inequality (2.5) again and by choosing $R_{1}=R_{0}+1$ and $t=R/2$ this times, we infer

[TABLE]

Observe that the second term on the right hand side is bounded by (2.6), we have

[TABLE]

Therefore, for $R\geq R_{0}$, the claim is prove.

**Step 3: **In this step, we will complete the proof of Lemma 2.1 by using (2.4)

[TABLE]

Indeed, letting $t=2$ and $R_{1}=R-4$ in (2.5), we obtain

[TABLE]

Thanks to (2.4), the first term of the right hand side is estimated by

[TABLE]

for $R-4\geq 2(R_{0}+1)$. Hence, by renaming $R$, the above inequality can be rewritten as

[TABLE]

Iterating this inequality $k$ times, we conclude that

[TABLE]

However, using (2.4) again, we deduce that the second term is bounded by

[TABLE]

which tends to [math] as $k\to\infty$. This implies

[TABLE]

for some constant $C>0$ independent of $R$, and the lemma follows. ∎

Remark 2.2*.*

Recently, in [MSW17], Munteanu et. all introduced a parabolic version of decay estimate for weighted volume and used it to investigate Poisson equation on complete smooth metric measure spaces.

Corollary 2.3**.**

Let $E$ be an end of a complete smooth metric measure space $(M,ds^{2}_{\rho},e^{-f}dv)$. Suppose that $\lambda_{1,f}(E)>0$, i. e.,

[TABLE]

for any compactly supported function $\varphi\in\mathcal{C}_{0}^{\infty}(E)$. Let $u$ be a nonnegative function defined on $E$ such that

[TABLE]

for some constant $\mu$ satisfying $\lambda_{1,f}-\mu>0$. If $u$ has the growth condition

[TABLE]

when $R\to\infty$, where $a=\sqrt[]{\lambda_{1,f}-\mu}$, then $u$ must satisfy the decay estimate

[TABLE]

for some constant $C>0$ depending on $f,$ and $a$.

Proof of Corollary 2.3.

By variational principle for $\lambda_{1,f}$, we have

[TABLE]

Let $\rho=a^{2}$, the distance function with respect to the complete metric $\rho ds^{2}$ is given by

[TABLE]

Now, we can apply Lemma 2.1 to complete the proof. ∎

Note that if $u\equiv 1$, we obtain the following decay estimate

Corollary 2.4**.**

Suppose that $E$ is a $f$-parabolic end of smooth metric measure space $(M,g,e^{-f}dv)$ with $\lambda_{1,f}(E)>0$. Denote by $V_{E,f}(R)$ the weighted volume of $E(R)$, then the following decay estimate

[TABLE]

holds true. Here $R_{0}$ is a given positive number and $C(\lambda_{1,f}(E))$ is some positive constant depending on $\lambda_{1,f}(E)$.

It is worth to mention that in [BK06], Buckley and Koskela also proved earlier a version of decay estimate for $f$-volume in a more general setting.

3. Counting cusps on smooth metric measure spaces of finite $f$-volume

As what we mentioned in the introduction part, in order to estimate the number of cusps we must have a decay estimate of the $f$-volume proved in Section 2 and a volume comparison theorem. Hence, first we introduce a volume comparison result given by Munteanu and Wang in [MW12].

Lemma 3.1**.**

Let $(M,g,e^{-f}dv)$ be a complete smooth metric measure space with $Ric_{f}\geq-(n-1)$. Suppose the weighted function has sublinear growth, i. e. for $x\in M$,

[TABLE]

for some nonnegative constants $\alpha$ and $\beta$. Then there exists a constant $C>0$ such that the volume upper bound

[TABLE]

holds for all $R>0$. Here $V_{f}(B(o,R))$ stands for the weighted volume of $B(o,R)$

Now, we will combine Lemma 3.1 and Corollary 2.4 to count number of cusps. The first result is as follows.

Theorem 3.2**.**

Let $(M,g,e^{-f}dv)$ be a smooth metric measure space with $Ric_{f}\geq-(n-1)$. Suppose that the weighted function is of sublinear growth, namely, for all $x\in M$,

[TABLE]

for some nonnegative constants $\alpha$ and $\beta$. If $M$ has finite $f$-volume given by $V$, and

[TABLE]

for some $R_{0}>0$, then

[TABLE]

Here $C>0$ is a constant depending on $\lambda_{1,f}$.

Proof of Theorem 3.2.

To simple the notation, we denote $V_{o,f}(R)$ to be the weighted volume of the geodesic ball $B(o,R)$, then for all $R>2(R_{0}+1)$, we have

[TABLE]

Here we used Lemma 2.4.

On the other hand, if $y\in\partial B(o,R+1)$ then $B(o,1)\subset B(y,R+2)$. Hence, we use Lemma 3.1 to obtain

[TABLE]

Suppose that $M\setminus B(o,R)$ has $N(R)$ unbounded components, then there exist $N(R)$ number of points $\{y_{i}\in\partial B(o,R+1)\}$ such that $B({y_{i}},1)\cap B({y_{j}},1)=\oslash$ for $i\neq j$. In particular, applying (3.2) to each of the $y_{i}$ and combining with (3.1), we have

[TABLE]

This implies that

[TABLE]

Note that $N(R)$ is the number of ends of $M$ with respect to $B(o,R)$, letting $R\to\infty$, we complete the proof. ∎

Next, we will derive a weighted version of a Li-Wang’s result in [LW10] to estimate $\lambda_{1,f}(B_{o}(R))$ of a geodesic ball centered at $o$ with radius $R$ in terms of the weighted volume of the ball. It is worth to mention that as in [LW10], we do not require any curvature assumptions on $M$.

Lemma 3.3**.**

Let $(M,g,e^{f}dv)$ be a complete smooth metric measure space. Then for any $0<\delta<1,R>2$ and $o\in M$, we have

[TABLE]

Proof.

Observe that

[TABLE]

Hence, we may assume $\lambda_{1,f}(B_{o}(R))\geq\dfrac{4}{R^{2}}$.

To simple the notation, let us use $\lambda_{1,f}$ to denote $\lambda_{1,f}(B_{o}(R))$. By the variational characteristic of $\lambda_{1,f}(B_{o}(R))$, we have

[TABLE]

for any nonnegative Lipschitz function $\phi$ with support in $B_{o}(R)$. Consequently,

[TABLE]

In particular, for $R>2,$ we choose

[TABLE]

then $\phi=1$ on $B_{o}(1)$ since $R-\lambda_{1,f}^{1/2}\geq R/2>1$. Plugging $\phi$ in the above inequality, we obtain

[TABLE]

Here we used $\sqrt[]{\lambda_{1,f}}(R-r)\leq 1$ on $B_{o}(R)\setminus B_{o}(R-\lambda_{1,f}^{1/2})$ in the last inequality. Therefore,

[TABLE]

This implies

[TABLE]

The lemma follows by rewriting this inequality. ∎

Note that $\lambda_{1,f}(M)=\lim_{R\rightarrow\infty}\lambda_{1,f}(B_{o}(R))$. If we first let $R$ going to infinity and then $\delta$ going to $1$ in the estimate of Lemma 3.3, we have the following result.

Corollary 3.4**.**

Let $(M,g,e^{f}dv)$ be a complete smooth metric measure space. Assume for some nonnegative constants $\alpha$ and $\beta$,

[TABLE]

for $x\in M$ and $\lambda_{1,f}(M)$ its bottom spectrum. Then

[TABLE]

Now, we will give a proof of Theorem 1.2.

Proof of Theorem 1.2.

Note that $o\in M$ be a fixed point. For any $0<\delta<1,$ let

[TABLE]

Thanks to Lemma 3.3, we have

[TABLE]

On the other hand, by the variational principle, we have

[TABLE]

Hence, combining this inequality with assumption regarding to $\mu_{1}(M)$, we infer

[TABLE]

So Theorem 3.2 implies

[TABLE]

To finish the proof, we first choose

[TABLE]

then replace the value of $R_{0},\delta$ in the last inequality. So we are done. ∎

As we mentioned in the introduction part, two main ingredients in our proof of Theorem 1.2 are the decay estimate of the volume and the volume comparison result. Therefore, using Lemma 2.4 and the volume comparison theorems in [MW14] and [Lam08], we can count the number of cusps on the following two manifolds.

Theorem 3.5**.**

Let $(M,g,e^{f}dv)$ be a complete smooth metric measure space with $Ric_{f}\geq\lambda$. Assume for some nonnegative constants $\alpha$ and $\beta$,

[TABLE]

for all $x\in M$. If $M$ has finite $f$-volume given by $V$, and

[TABLE]

Let us denote $N(M)$ to be the number of ends (cusps) of $M$. Then there exists a contant $C(n)>0$ depending only on $n$, such that,

[TABLE]

where $V_{o}(1)$ denotes the (non-weighted) volume of the unit ball centered at any fixed point $o\in M$. Here $\mu_{1}(M)$ is defined by the (non-weighted) Reileigh quotient,

[TABLE]

Theorem 3.6**.**

Let $M$ be a complete noncompact $16$-dimensional manifold with holonomy group Spin(9) with $\mu_{1}(M)\geq 121$. Then there exists a contant $C(n)>0$ depending only on $n$, such that,

[TABLE]

where $V_{o}(1)$ denotes the (non-weighted) volume of the unit ball centered at any point $o\in M$. Here $\mu_{1}(M)$ is defined by the (non-weighted) Reileigh quotient,

[TABLE]

4. Counting cusps via the $p$-Laplacian

In this section, we will use the nonlinear theory of $p$-Laplacian to estimate the number of cusps of Riemannian manifolds. Again, our strategy is to use a (nonlinear) decay estimate of the volume and corresponding volume comparison theorem. Therefore, let us recall the following nonlinear version regarding to the rates of volume decay.

Theorem 4.1** ([BK06]).**

Let $E$ be an end of a complete Riemannian manifold $(M^{n},g)$ with respect to $\overline{B}(o,R_{0})$. Suppose that the first eigenvalues of the $p$-Laplacian $\lambda_{1,p}(E)>0,(1\leq p<\infty)$. If $E$ has finite volume then the following decay estimate

[TABLE]

for all $R>R_{0}+2$.

Proof.

We note that in the proof of the first part of Theorem 0.1 in [BK06], the author showed that for $R>R_{0}+2$, we have

[TABLE]

The proof is complete. ∎

We can estimate the number of cusps of smooth metric measure spaces as follows.

Theorem 4.2**.**

Let $(M^{n},g)$ be a smooth metric measure space with $Ric\geq-(n-1)$. If $M$ has finite volume given by $V$, and

[TABLE]

for some $R_{0}>0$, then

[TABLE]

Here $C>0$ is a constant depending on $\lambda_{1,p}$.

Proof.

By using the decay estimate in Theorem 4.1 and the volume comparison theorem 3.1, we can repeat the proof of Theorem 3.2 to derive the conclusion. Since they are almost the same, we omit the detail. ∎

Next, we will estimate $\lambda_{1,p}$ on the ball $B(o,R)$.

Lemma 4.3**.**

Let $(M^{n},g)$ be a Riemannian manifold. Then for any $0<\delta<1,1\leq p<\infty$ and $o\in M$, we have

[TABLE]

where $\alpha(p)=\max\{1,p/2\}$.

Proof.

Observe that

[TABLE]

Hence, we may assume $\lambda_{1,p}(B_{o}(R))\geq\dfrac{2^{p}}{R^{p}}$.

To simple the notation, let us use $\lambda_{1,p}$ to denote $\lambda_{1,p}(B_{o}(R))$. By the variational characteristic of $\lambda_{1,p}(B_{o}(R))$, we have

[TABLE]

for any nonnegative Lipschitz function $\phi$ with support in $B_{o}(R)$. We have two cases.

Case 1: $p\geq 2$. Observe that $x^{p/2}$ is a convex function, we have the following basic inequality

[TABLE]

for any $A,B\geq 0$. Hence, by (4.1), we have

[TABLE]

Consequently,

[TABLE]

Here we used $0<\delta<1$ in the last inequality.

Case 2: $1\leq p<2$. Since $0<p/2<1$, we have for $A,B\geq 0$

[TABLE]

Therefore, by (4.1), we obtain

[TABLE]

Since $0<\delta<1$, this implies,

[TABLE]

In conclusion, we obtain in both cases that

[TABLE]

Now, for $R>2,$ we choose

[TABLE]

then $\phi=1$ on $B_{0}(1)$ since $R-\lambda_{1,p}^{-1/p}\geq\frac{R}{2}>1$. Plugging $\phi$ in the above inequality, we obtain

[TABLE]

Here we used $0\leq\phi\leq 1$ in the third inequality. Therefore,

[TABLE]

This implies

[TABLE]

The lemma follows by rewriting this inequality. ∎

Corollary 4.4**.**

Let $M$ be a complete smooth metric measure spaces and $\lambda_{1,p}$ is its weighted $p$-spectrum. Then

[TABLE]

Proof.

We will use Lemma 4.3 to give the proof. First, we let $R\to\infty$ then let $\delta\to 1$, we obtain the conclusion. Hence, the proof is complete. ∎

Now we give a proof of Theorem 1.3.

Proof of Theorem 1.3.

Note that $o\in M$ be a fixed point. Let

[TABLE]

Thanks to Lemma 4.3, we infer

[TABLE]

On the other hand, by the variational principle, we have

[TABLE]

Hence, combining this inequality with assumption regarding to $\mu_{1,p}(M)$, we obtain

[TABLE]

So Theorem 4.2 implies

[TABLE]

To finish the proof, we first choose $\delta=1-\dfrac{1}{\ln(V.V_{o}(1)^{-1})}$ then replace the value of $R_{0},\delta$ in the last inequality. So we are done. ∎

Similarly, using the volume comparison theorems in [Lam08, KLZ08, LW05], we have the following theorems.

Theorem 4.5**.**

Let $M^{m}$ be a complete Kähler manifold of complex dimension $m$ with finite volume. Assume that $M$ has holomorphic bisectional curvatures satisfying

[TABLE]

for all unitary frame $\{e_{1},\ldots,e_{m}\}$. If

[TABLE]

then there exists a contant $C(n,p)>0$ depending only on $n$ and $p$ such that,

[TABLE]

where $\alpha(p)=\max\{1,p/2\}$ and $V_{o}(1)$ denotes the volume of the unit ball centered at any point $o\in M$

Theorem 4.6**.**

Let $(M^{n},g)$ be a complete quarternionic Kähler of real dimension $4m$ with finite volume given by $V$. Assume that its scalar curvature of satisfies the bound

[TABLE]

and

[TABLE]

Let us denote $N(M)$ to be the number of ends (cusps) of $M$. Then there exists a contant $C(n,p)>0$ depending only on $n$ and $p$ such that,

[TABLE]

where $\alpha(p)=\max\{1,p/2\}$ and $V_{o}(1)$ denotes the volume of the unit ball centered at a fixed point $o\in M$.

Theorem 4.7**.**

Let $(M^{n},g)$ be a Let $M$ be a complete noncompact $16$-dimensional manifold with holonomy group Spin(9) with finite volume given by $V$. Assume that

[TABLE]

Let us denote $N(M)$ to be the number of ends (cusps) of $M$. Then there exists a contant $C(n,p)>0$ depending only on $n$ and $p$ such that,

[TABLE]

where $\alpha(p)=\max\{1,p/2\}$ and $V_{o}(1)$ denotes the volume of the unit ball centered at a fixed point $o\in M$.

Acknowlegement

A part of this paper was written during a stay of the first author and the third author at the Vietnam Institute for Advanced Study in Mathematics (VIASM). They would like to express their thanks to the staff there for the hospitality and support.