On the spectrum of the Laplacian

Abstract

In this article we prove a generalization of Weyl’s criterion for the essential spectrum of a self-adjoint operator on a Hilbert space. We then apply this criterion to the Laplacian on functions over open manifolds and get new results for its essential spectrum.

1 Introduction

Let $M$ be a complete noncompact Riemannian manifold of dimension $n$ and denote by $\Delta$ the Laplacian acting on ${\mathcal {C}}_0^\infty (M)$. It is well known that the self-adjoint extension of $\Delta$ on $L^2(M)$ exists and is a unique nonpositive definite and densely defined linear operator. We will also use $\Delta$ to denote this extension for the remaining paper.

The spectrum of $-\Delta ,\,\sigma (-\Delta )$, consists of all points $\lambda \in {\mathbb {C}}$ for which $\Delta +\lambda I$ fails to be invertible. Since $-\Delta$ is nonnegative definite, its $L^2$-spectrum is contained in $[0,\infty )$. The essential spectrum of $-\Delta ,\,\sigma _ess (-\Delta )$, consists of the cluster points in the spectrum and of isolated eigenvalues of infinite multiplicity. The following result is due to Donnelly [5]: if there exists an infinite dimensional subspace $G$ in the domain of $\Delta$ such that

$$\begin{aligned} \Vert \Delta u+\lambda u\Vert _{L^2}\le \sigma \Vert u\Vert _{L^2} \end{aligned}$$

(1)

for all $u\in G$, then

$$\begin{aligned} \sigma _\mathrm {ess}(-\Delta )\cap (\lambda -\sigma ,\lambda +\sigma )\ne \emptyset . \end{aligned}$$

The functions $u$ are referred to as the approximate eigenfunctions corresponding to the eigenvalue $\lambda$. The above criterion is simple to apply and has directed the study of the essential spectrum of the Laplacian for the last three decades. A related result of the above is as follows: let $u$ be a nonzero smooth function with compact support. If (1) is satisfied, then

$$\begin{aligned} \sigma (-\Delta )\cap (\lambda -\sigma ,\lambda +\sigma )\ne \emptyset . \end{aligned}$$

We remark that for the above criteria to be valid, we do not have to assume the completeness of the manifold $M$. They can be applied to both compact and noncompact manifolds with boundary (with either Dirichlet or Neumann boundary conditions) as well as complete compact and noncompact manifolds. If $M$ is compact, the criterion gives the eigenvalue estimates.

In most problems the ideal space to work on is the $L^2$ function space when compared to the $L^q$ spaces. However, this is not the case when considering the spectrum of the Laplacian. On a Riemannian manifold, most of the approximate eigenfunctions we can write out explicitly must be related to the distance function. It is well known however, that the Laplacian of the distance function is locally bounded in $L^1$, but not in $L^2$.

We can see this in the following simple example. Take $M=S^1\times (-\infty ,\infty )$, letting $(\theta ,x)$ be the coordinates. Then the radial function with respect to the point (0,0) is given by

$$\begin{aligned} r=\sqrt{x^2+(\min (\theta , 2\pi -\theta ))^2}. \end{aligned}$$

A straightforward computation gives

$$\begin{aligned} \Delta r=-\frac{2\pi }{\sqrt{x^2+\pi ^2}}\delta _{\{\theta =\pi \}}+\hbox { a smooth function}, \end{aligned}$$

where $\delta _{\{\theta =\pi \}}$ is the Delta function along the submanifold $\{\theta =\pi \}$. Therefore $\Delta r$ is not locally $L^2$.

The failure of the $L^2$ integrability of the Laplacian of the distance function was one of the main difficulties in applying the classical criterion above. In fact, it was not possible to prove that the $L^2$ essential spectrum of the Laplacian on a manifold with nonnegative Ricci curvature is $[0,\infty )$ by directly computing the $L^2$ spectrum. Additional assumptions on the curvature and geometry of the manifold were necessary (see for example [2, 6, 9, 10, 13, 20]).

Donnelly [6] proved that the essential spectrum of the Laplacian is $[0,\infty )$ for manifolds of nonnegative Ricci curvature and maximal volume growth. Wang [19], by using the seminal theorem of Sturm [18], removed the maximal volume growth condition. Wang’s result confirms the conjecture that the spectrum of manifolds with nonnegative Ricci curvature is $[0,\infty )$. In [16], Lu-Zhou gave a technical generalization of Wang’s result which includes the case of manifolds of finite volume.

In this article, we introduce a new method for computing the spectrum of a self-adjoint operator on a Hilbert space (see Theorem 2.3) which has the following application in the case of the Laplacian:

Theorem 1.1

Let $M$ be a Riemannian manifold and let $\Delta$ be the Laplacian. Assume that for $\lambda \in {\mathbb {R}}^+$, there exists a nonzero function $u$ in the domain of $\Delta$ such that

$$\begin{aligned} \Vert u\Vert _{L^\infty }\cdot \Vert \Delta u+\lambda u\Vert _{L^1}\le \delta \Vert u\Vert _{L^2}^2 \end{aligned}$$

(2)

for some positive number $\delta >0$. Then

$$\begin{aligned} \sigma (-\Delta )\cap (\lambda -\varepsilon ,\lambda +\varepsilon )\ne \emptyset , \end{aligned}$$

where

$$\begin{aligned} \varepsilon =\min (1,(\lambda +2)\delta ^{1/3}). \end{aligned}$$

Moreover,

$$\begin{aligned} \sigma _{ess } (-\Delta )\cap (\lambda -\varepsilon ,\lambda +\varepsilon )\ne \emptyset , \end{aligned}$$

if for any compact subset $K$ of $M$, there exists a nonzero function $u$ in the domain of $\Delta$ satisfying  (2) whose support is outside $K$.

We expect the above result to have many applications in spectrum theory (for example, on manifolds with corners, where the test functions are usually not smooth). In this paper, we concentrate on applying the criterion to the computation of the essential spectrum of complete noncompact manifolds.

Theorem 1.1 proves to be a powerful tool in expanding the set of manifolds for which the $L^2$ essential spectrum is the nonnegative real line. In the case of shrinking Ricci solitons, we are able to prove the following result.

Theorem 1.2

The $L^2$ essential spectrum of a complete shrinking Ricci soliton is $[0,\infty )$.

Note that no curvature assumption is needed here.

For a large class of manifolds (for example, the shrinking Ricci solitons), we are able to control the volume growth about a fixed point, but it is difficult to prove the uniformness of their volume growth. Without the uniform volume growth property, the theorem of Sturm does not apply and the results of Wang [19] and Lu-Zhou [16] cannot be used. Therefore, the following result may be practically useful:

Theorem 1.3

Let $M$ be a complete noncompact Riemannian manifold. Suppose that, with respect to a fixed point $p$, the radial Ricci curvature is asymptotically nonnegative (see Lemma 5.2). If the volume of the manifold is finite we additionally assume that its volume does not decay exponentially at $p$. Then the $L^2$ spectrum of the Laplace operator on functions is $[0,\infty )$.

We shall also use Theorem 1.1 to modify a result of Elworthy-Wang [8] on manifolds that posses an exhaustion function (Theorem 8.2). We replace the $L^2$ norm assumption by an $L^1$ norm assumption.

In the last section, we show that it is possible to work with continuous test functions in Theorem 1.1. By using them instead we avoid the repetitive choosing of cut-off functions.

The essential spectrum of the Laplacian on noncompact Riemannian manifolds is interesting and important as it reveals a lot of information about the geometry of the manifold. Although there are lot of interesting open problems in this direction, the authors believe that answering the following conjecture is the most important one.

Conjecture 1.4

Let $M$ be a complete noncompact Riemannian manifold with Ricci curvature bounded below. Then the $L^2$ essential spectrum of the Laplacian on functions is a connected subset of the real line. In other words, the essential spectrum set is of the form $[a,\infty )$, where $a$ is a nonnegative real number.

As is well-known, the essential spectrum of a Schrödinger operator could be very complicated (cf. [17, Chapter XIII]) and it certainly need not be a connected set. For the case of the Laplacian on a complete manifold however, in all known examples the $L^2$ essential spectrum is a connected set. In this paper, we answer the above conjecture in some special cases. We believe that the analysis of the wave kernel is needed to answer the conjecture in full.

2 The Weyl criterion for quadratic forms

Let $H$ be a self-adjoint operator on a Hilbert space ${\mathcal {H}}$. The norm and inner product in ${\mathcal {H}}$ are respectively denoted by $\Vert \cdot \Vert$ and $(\cdot ,\cdot )$. Let $\sigma (H), \sigma _\mathrm {ess}(H)$ be the spectrum and the essential spectrum of $H$, respectively. Let ${\mathfrak {D}}(H)$ be the domain of $H$. The Classical Weyl criterion states that

Theorem 2.1

(Classical Weyl’s criterion) A nonnegative real number $\lambda$ belongs to $\sigma (H)$ if, and only if, there exists a sequence $\{\psi _n\}_{n \in {\mathbb {N}}} \subset {\mathfrak {D}}(H)$ such that

• (1) $\forall n\in {\mathbb {N}}, \quad \Vert \psi _n\Vert =1$ ,

• (2) $(H-\lambda )\psi _n \rightarrow 0, \hbox { as } n\rightarrow \infty \hbox { in }\mathcal {H}.$

Moreover, $\lambda$ belongs to $\sigma _\mathrm {ess}(H)$ of $H$ if, and only if, in addition to the above properties

• (3) $\psi _n \rightarrow 0 \hbox { weakly as } n\rightarrow \infty \hbox { in }\mathcal {H}.$

Remark 2.2

The above theorem is still true if the convergence in (2) is replaced by weak convergence, the statement of which can be found (without proof) in [4] and later in [15]. This version of the Weyl criterion was applied for the first time to the Laplacian on curved Euclidean domains in [15]. The authors are grateful to David Krejčiřík for informing them of the results.

We have the following functional analytic result, which generalizes the weak Weyl criterion. To the authors’ knowledge, this result seems to be new.

Theorem 2.3

Let $f$ be a bounded positive continuous function over $[0,\infty )$. A nonnegative real number $\lambda$ belongs to the spectrum $\sigma (H)$ if, and only if, there exists a sequence $\{\psi _n\}_{n \in {\mathbb {N}}} \subset {\mathfrak {D}}(H)$ such that

• (1) $\forall n\in {\mathbb {N}}, \quad \Vert \psi _n\Vert =1$ ,

• (2) $(f(H) (H-\lambda )\psi _n, (H-\lambda )\psi _n)\rightarrow 0, \hbox { as } n\rightarrow \infty \quad {and}$

• (3) $(\psi _n, (H-\lambda )\psi _n) \rightarrow 0, \hbox { as } n\rightarrow \infty .$

Moreover, $\lambda$ belongs to $\sigma _\mathrm {ess}(H)$ of $H$ if, and only if, in addition to the above properties

• (4) $\psi _n \rightarrow 0, \hbox { weakly as } n\rightarrow \infty$ in $\mathcal {H}$.

Proof

Since $H$ is a densely defined self-adjoint operator, the spectral measure $E$ exists and we can write

$$\begin{aligned} H=\int _0^\infty \lambda \, dE. \end{aligned}$$

(3)

Assume that $\lambda \in \sigma (H)$. Then by Weyl’s criterion, there exists a sequence $\{\psi _n\}$ such that

$$\begin{aligned} \Vert (H-\lambda )\psi _n\Vert \rightarrow 0, \quad \Vert \psi _n\Vert =1 \end{aligned}$$

as $n\rightarrow \infty$.

We write

$$\begin{aligned} \psi _n=\int _0^\infty d E(t)\psi _n \end{aligned}$$

as its spectral decomposition. Then

$$\begin{aligned} (f(H) (H-\lambda )\psi _n, (H-\lambda )\psi _n)=\int _0^\infty f(t)(t-\lambda )^2 d\Vert E(t)\psi _n\Vert ^2. \end{aligned}$$

Since $f$ is a bounded positive function, we have

$$\begin{aligned} (f(H) (H-\lambda )\psi _n, (H-\lambda )\psi _n)\le C\int _0^\infty (t-\lambda )^2 d\Vert E(t)\psi _n\Vert ^2=C\Vert (H-\lambda )\psi _n\Vert ^2. \end{aligned}$$

Moreover,

$$\begin{aligned} (\psi _n, (H-\lambda )\psi _n)\le C\,\Vert \psi _n\Vert \cdot \Vert (H-\lambda )\psi _n\Vert . \end{aligned}$$

Thus the necessary part of the theorem is proved.

Now assume that $\lambda >0$ and $\lambda \notin \sigma (H)$. Then there is a $\lambda >\varepsilon >0$ such that $E(\lambda +\varepsilon )-E(\lambda -\varepsilon )=0$. We write

$$\begin{aligned} \psi _n=\psi _n^1+\psi _n^2, \end{aligned}$$

(4)

where

$$\begin{aligned} \psi _n^1=\int _0^{\lambda -\varepsilon } dE(t)\psi _n, \end{aligned}$$

and $\psi _n^2=\psi _n-\psi _n^1$.

Then

$$\begin{aligned}&(f(H) (H-\lambda )\psi _n, (H-\lambda )\psi _n) \\&\quad =\left( f(H) (H-\lambda )\psi _n^1, (H-\lambda )\psi _n^1\right) +\left( f(H) (H-\lambda )\psi ^2_n, (H-\lambda )\psi _n^2\right) \\&\quad \ge c_1\Vert \psi _n^1\Vert ^2+(f(H) (H-\lambda )\psi ^2_n, (H-\lambda )\psi _n^2)\ge c_1\Vert \psi _n^1\Vert ^2, \end{aligned}$$

where the positive number $c_1$ is the infimum of the function $f(t)(t-\lambda )^2$ on $[0,\lambda -\varepsilon ]$. Therefore

$$\begin{aligned} \Vert \psi _n^1\Vert \rightarrow 0 \end{aligned}$$

by (2). On the other hand, we similarly get

$$\begin{aligned} (\psi _n, (H-\lambda )\psi _n)\ge \varepsilon \Vert \psi _n^2\Vert ^2-\lambda \Vert \psi _n^1\Vert ^2. \end{aligned}$$

If the criteria (2), (3) are satisfied, then, by the two estimates above, we conclude that both $\psi _n^1, \psi _n^2$ go to zero. This contradicts $\Vert \psi _n\Vert =1$, and the theorem is proved.

Note that for $\lambda =0,\,\psi _n^1$ is automatically zero, and the second half of the proof would give the contradiction. $\square$

We apply Theorem 2.3 to the Laplacian on functions. In this setting two particular cases of the function $f$ will be of interest.

Corollary 2.1

A nonnegative real number $\lambda$ belongs to the spectrum $\sigma (H)$ if, and only if, there exists a sequence $\{\psi _n\}_{n \in {\mathbb {N}}} \subset {\mathfrak {D}}(H)$ such that

• (1) $\forall n\in {\mathbb {N}}, \quad \Vert \psi _n\Vert =1$ ,

• (2) $((H+1)^{-1}\psi _n, (H-\lambda )\psi _n)\rightarrow 0, \hbox { as } n\rightarrow \infty \quad {and}$

• (3) $(\psi _n, (H-\lambda )\psi _n) \rightarrow 0, \hbox { as } n\rightarrow \infty .$

Moreover, $\lambda$ belongs to $\sigma _\mathrm {ess}(H)$ of $H$ if, and only if, in addition to the above properties

• (4) $\psi _n \rightarrow 0, \hbox { weakly as } n\rightarrow \infty$ in $\mathcal {H}$.

Proof

We take $f(x)=(x+1)^{-1}$. The corollary follows from the identity

$$\begin{aligned} (H+1)^{-1}(H-\lambda )=1-(\lambda +1)(H+1)^{-1}. \end{aligned}$$

$\square$

In a similar spirit, taking $f(x)=(x+\alpha )^{-(N+1)}$ for a natural number $N$ and a positive number $\alpha >1$, we also obtain the following generalization

Corollary 2.2

A nonnegative real number $\lambda$ belongs to the spectrum $\sigma (H)$ if, and only if, there exists a sequence $\{\psi _n\}_{n \in {\mathbb {N}}} \subset {\mathfrak {D}}(H)$ such that

• (1) $\forall n\in {\mathbb {N}}, \quad \Vert \psi _n\Vert =1$ ,

• (2) $((H+\alpha )^{-i}\psi _n, (H-\lambda )\psi _n)\rightarrow 0 \hbox { as } n\rightarrow \infty \$ for two consecutive natural numbers i=N, N+1, and

• (3) $(\psi _n, (H-\lambda )\psi _n) \rightarrow 0, \hbox { as } n\rightarrow \infty .$

Moreover, $\lambda$ belongs to $\sigma _\mathrm {ess}(H)$ of $H$ if, and only if, in addition to the above properties

• (4) $\psi _n \rightarrow 0, \hbox { weakly as } n\rightarrow \infty$ in $\mathcal {H}$.

Remark 2.4

Using the Cauchy inequality, the above two corollaries reduce to Donnelly’s criterion (1) when we consider the case $H=-\Delta$.

3 A spectrum estimate result

In this section we will prove a special version of Theorem 2.3 for the Laplacian on functions. We begin with the fact that its resolvent is always bounded on $L^\infty$.

Lemma 3.1

We have

$$\begin{aligned} (-\Delta +1)^{-1} \end{aligned}$$

is bounded from $L^\infty (M)$ to itself and the operator norm is no more than 1.

The lemma follows from the proof of Lemma 3.1 in [1]. The resolvent is bounded on $L^\infty$ because the heat kernel is bounded on $L^\infty$. This is a property that Davies proves for any nonnegative self-adjoint operator that satisfies Kato’s inequality like the Laplacian [3, Theorems 1.3.2, 1.3.3]. It is due to the well-known fact that the Laplacian on functions is a self-adjoint operator that satisfies Kato’s inequality. Together with Corollary 2.1 this lemma allows us to obtain an even simpler criterion for the essential spectrum of the Laplacian on functions:

Proof of Theorem 1.1

Assume that

$$\begin{aligned} \sigma (-\Delta )\cap (\lambda -\varepsilon ,\lambda +\varepsilon )=\emptyset . \end{aligned}$$

By the above lemma, we have

$$\begin{aligned}&|(u, (-\Delta -\lambda )u)|\le \Vert u\Vert _{L^\infty }\cdot \Vert (-\Delta -\lambda )u\Vert _{L^1}\\&|((-\Delta +1)^{-1}(-\Delta -\lambda )u, (-\Delta -\lambda )u)|\le (\lambda +2)\Vert u\Vert _{L^\infty }\cdot \Vert (-\Delta -\lambda )u\Vert _{L^1} \end{aligned}$$

We write

$$\begin{aligned} u=u_1+u_2 \end{aligned}$$

according to the spectral decomposition of the operator $-\Delta$ (cf.  4). Then we have

$$\begin{aligned}&\varepsilon \Vert u_2\Vert _{L^2}^2-\lambda \Vert u_1\Vert _{L^2}^2\le \delta \Vert u\Vert ^2_{L^2};\\&\Vert u_1\Vert _{L^2}^2\le \frac{(\lambda +2)(\lambda +1)}{\varepsilon ^2}\delta \Vert u\Vert _{L^2}^2. \end{aligned}$$

Thus we have

$$\begin{aligned} \varepsilon \Vert u_2\Vert _{L^2}^2+\varepsilon \Vert u_1\Vert _{L^2}^2\le \left( \frac{(\lambda +2)(\lambda +1)(\lambda +\varepsilon )}{\varepsilon ^2}+1\right) \delta \Vert u\Vert _{L^2}^2. \end{aligned}$$

Since $\Vert u_2\Vert _{L^2}^2+\Vert u_1\Vert _{L^2}^2=\Vert u\Vert _{L^2}^2$, we get

$$\begin{aligned} \varepsilon \le \left( \frac{(\lambda +2)(\lambda +1)(\lambda +\varepsilon )}{\varepsilon ^2}+1\right) \delta \end{aligned}$$

which is a contradiction. The essential spectrum result of the theorem follows from the classical Weyl criterion (Theorem 2.1, (3)). $\square$

4 An approximation theorem

Let $M$ be a complete noncompact Riemannian manifold. Let $p\in M$ be a fixed point. Define

$$\begin{aligned} r(x)=d(x,p) \end{aligned}$$

to be the radial function on $M$. It is well known that

• (1) $r(x)$ is continuous;

• (2) $|\nabla r(x)|=1$ almost everywhere and $r(x)$ is a Lipschitz function;

• (3) $\Delta r$ exists on $M\backslash \{p\}$ in the sense of distribution.

In general, it is not possible to find smooth approximations of a Lipschitz function in the ${\mathcal {C}}^1$ norm. The following Proposition, which is a more precise version of [16, Proposition 1], implies that this can be done up to a function with small $L^1$ norm. Such kind of result is essential in Riemannian geometry and should be well-known, but given that we were not able to find a reference, we include a proof.

Proposition 4.1

For any positive continuous decreasing function $\eta : {\mathbb {R}}^+\rightarrow {\mathbb {R}}^+$ such that

$$\begin{aligned} \lim _{r\rightarrow \infty } \eta (r)=0, \end{aligned}$$

there exist $C^\infty$ functions $\tilde{r}(x)$ and $b(x)$ on $M$ such that

• (a) $\Vert b\Vert _{L^1(M\backslash B_{{p}}(R))}\le \eta (R-1)$;

• (b) $\Vert \nabla \tilde{r}-\nabla r\Vert _{L^1(M\backslash B_{{p}}(R))}\le \eta (R)$

and for any $x\in M$ with $r(x)>2$

• (c) $|\tilde{r}(x)-r(x)|\le \eta (r(x))$ and $|\nabla \tilde{r} (x)|\le 2$;

• (d) $\Delta \tilde{r}(x)\le \max _{y\in B_x(1)} \Delta r(y)+\eta (r(x))+|b(x)|$ in the sense of distribution.

Proof

Without loss of generality, we assume that $\eta (r)<1$. Let $\{U_i\}$ be a locally finite cover of $M$ and let $\{\psi _i\}$ be the partition of unity subordinate to the cover. Let $\mathbf{x_i}=(x_i^1,\ldots ,x_i^n)$ be the local coordinates of $U_i$. Define $r_i=r|_{U_i}$.

Let $\xi (\mathbf x)$ be a non-negative smooth function on ${\mathbb {R}}^n$ whose support is within the unit ball. Assume that

$$\begin{aligned} \int _{{\mathbb {R}}^n}\xi =1. \end{aligned}$$

Without loss of generality, we assume that each $U_i$ is an open subset of the unit ball of ${\mathbb {R}}^n$ with coordinates $\mathbf{x_i}$. Then for any $\varepsilon >0$,

$$\begin{aligned} r_{i,\varepsilon }=\frac{1}{\varepsilon ^n}\int _{{\mathbb {R}}^n}\xi \left( \frac{\mathbf{x_i}-\mathbf{y_i}}{\varepsilon }\right) r_i(\mathbf{y_i}) d\mathbf{y_i} \end{aligned}$$

is a smooth function on $U_i$ and hence on $M$. Let $\{\sigma _i\}$ be a sequence of positive numbers such that

$$\begin{aligned} \sum _i\sigma _i(|\Delta \psi _i(x)|+4|\nabla \psi _i (x)|+\psi _i(x))\le \eta (r(x)). \end{aligned}$$

(5)

By [11, Lemma 7.1, 7.2], for each $i$, we can choose $\varepsilon _i<1$ small enough so that

$$\begin{aligned} \begin{aligned}&|r_{i,\varepsilon _i}(x)-r_i(x)|\le \sigma _i;\\&\Vert \nabla r_{i,\varepsilon _i}-\nabla r_i\Vert _{L^1{(U_i)}}\le \sigma _i. \end{aligned} \end{aligned}$$

(6)

We also have

$$\begin{aligned} \Delta r_{i,\varepsilon _i}(x)\le \max _{y\in B_x(1)}\Delta r_i(y). \end{aligned}$$

(7)

Define

$$\begin{aligned} \tilde{r}=\sum _i\psi _i r_{i,\varepsilon _i}, \quad b=2 { \sum _i} \nabla \psi _i\cdot \nabla r_{i,\varepsilon _i}. \end{aligned}$$

Since $\sum _i (\nabla \psi _i\cdot \nabla r_i)=(\sum _i\nabla \psi _i) \cdot \nabla r=0$ almost everywhere on $M$, we have

$$\begin{aligned} b=2\sum _i \nabla \psi _i\cdot (\nabla r_{i,\varepsilon _i}-\nabla r_i) \end{aligned}$$

almost everywhere. Thus (a) follows. Similarly, observing that

$$\begin{aligned} \tilde{r}-r=\sum _i\psi _i(r_{i,\varepsilon _i}-r_i), {\quad \hbox {and}\ \ |\nabla r_{i,\varepsilon _i}|<2,} \end{aligned}$$

we obtain (b), (c).

To prove (d), we compute

$$\begin{aligned} \Delta \tilde{r}=\sum _i[(\Delta \psi _i)\, r_{i,\varepsilon _i}+2\nabla \psi _i\nabla r_{i,\varepsilon _i}+\psi _i\Delta r_{i,\varepsilon _i}], \end{aligned}$$

and since

$$\begin{aligned} \sum _i(\Delta \psi _i ) r_i=\sum _i (\Delta \psi _i) r=0, \end{aligned}$$

we have

$$\begin{aligned} \Delta \tilde{r}=\sum _i[\Delta \psi _i(r_{i,\varepsilon _i}- r_i)+b+\psi _i\Delta r_{i,\varepsilon _i}]. \end{aligned}$$

By (7), we obtain (d) and the Proposition is proved. $\square$

5 Manifolds with $\Delta r$ asymptotically nonpositive

As we have mentioned in the previous section, the Laplacian of the radial function $r(x)=d(x,p)$ exists in the sense of distribution (except at $p$). That is, for any nonnegative smooth function $f$ with compact support in $M\backslash \{p\}$, the integral

$$\begin{aligned} \int _Mf \Delta r \end{aligned}$$

is defined. The following simple observation is due to Wang [19] and is crucial in our estimates.

Lemma 5.1

The function $\Delta r$ is locally integrable away from $p$.

Proof

Let $W$ be any compact set of the form $B_p(R)-B_p(r)$ for $R>r>0$. Then by the Laplacian comparison theorem, there is a constant $C$, depending only on the dimension, $r,\,R$, and the lower bound of the Ricci curvature on $B_p(R)$, such that

$$\begin{aligned} \Delta r\le C \end{aligned}$$

on $W$ in the sense of distribution. Thus we have

$$\begin{aligned} |\Delta r|=|C-\Delta r-C|\le 2C-\Delta r \end{aligned}$$

and therefore

$$\begin{aligned} \int _W|\Delta r|\le 2C\,\mathrm{vol}\,(W) -\int _W\Delta r. \end{aligned}$$

Using Stokes’ Theorem, we obtain

$$\begin{aligned} \int _W |\Delta r|\,\le 2C\,\mathrm{vol}\,(W)-\int _{\partial W}\frac{\partial r}{\partial n}\le 2C\mathrm{vol}\,(W)+\mathrm{vol}\,(\partial W), \end{aligned}$$

and the lemma is proved. $\square$

In this section, we study manifolds with the following property

$$\begin{aligned} \overline{\lim _{r\rightarrow \infty }}\; \Delta r\le 0 \end{aligned}$$

(8)

in the sense of distribution, where $r(x)$ is the radial distance of $x$ to a fixed point $p$. We shall give a precise estimate of the $L^1$ norm of $\Delta r$ in terms of the volume growth of the manifold. But before we do that, we first provide an important example where the above technical condition holds.

We note that for a fixed point $p\in M$ the cut locus $\hbox {Cut}(p)$ is a set of measure zero in $M$. The manifold can be written as the disjoint union $M=\Omega \cup \hbox { Cut}(p)$, where $\Omega$ is star-shaped with respect to $p$. That is, if $x\in \Omega$, then the geodesic line segment $\overline{px}\subset \Omega$. $\partial r= \partial /\partial r$ is well defined on $\Omega$. We have the following result:

Lemma 5.2.

Let $r(x)$ be the radial function with respect to $p$. Suppose that there exists a continuous function $\delta (r)$ on ${\mathbb {R}}^+$ such that

• (i) $\lim _{r\rightarrow \infty } \delta (r)=0$;

• (ii) $\delta (r)>0$ and

• (iii) $\hbox {Ric}(\partial r, \partial r)\ge -(n-1) \delta (r)$ on $\Omega$.

Then (8) is valid in the sense of distribution.

Proof

On $\Omega$, we have the following Bochner formula

$$\begin{aligned} 0=\frac{1}{2}\Delta |\nabla r|^2=|\nabla ^2r |^2+\nabla r\cdot \nabla (\Delta r)+\hbox {Ric}(\partial r,\partial r). \end{aligned}$$

Since $\nabla ^2 r(\partial r, \partial r)=0$, using the Cauchy inequality, we have

$$\begin{aligned} 0\ge \frac{1}{n-1}(\Delta r)^2+\frac{\partial }{\partial r}(\Delta r) -(n-1)\delta (r). \end{aligned}$$

(9)

Given that $\Omega$ is star-shaped, for any fixed direction $\partial /\partial r$, we obtain (8) by comparing the above differential inequality with the Riccati equation.

On the points where $r$ is not smooth, we may use the trick of Gromov as in Proposition 1.1 of [14] to conclude the result in the sense of distribution. $\square$

5.1 Volume comparison theorems

Let $p$ be the fixed point of the manifold. Denote

$$\begin{aligned} B(r)=B_p(r),\quad V(r)=\mathrm{vol}\,(B_p(r)) \end{aligned}$$

the geodesic ball of radius $r$ at $p$ and its volume respectively.

The following volume comparison theorem is well-known.

Lemma 5.3

Let $r(x)$ be the radial function defined above. Assume that  (8) is valid in the sense of distribution. Then the manifold has subexponential volume growth at $p$. In other words, for all $\varepsilon >0$ there exists a positive constant $C(\varepsilon )$, depending only on $\varepsilon$ and the manifold, such that for all $R>0$

$$\begin{aligned} V(R)\le C(\varepsilon ) \, e^{\varepsilon \, R}. \end{aligned}$$

Proof

Let $m(r)$ be a nonnegative continuous function such that

$$\begin{aligned} \lim _{r\rightarrow \infty } m(r)=0, \end{aligned}$$

and

$$\begin{aligned} \Delta r\le m(r) \end{aligned}$$

in the sense of distribution. It follows that

$$\begin{aligned} \int _{B(R)\backslash B(1)}\Delta r\le \int _{B(R)\backslash B(1)} m(r) \end{aligned}$$

which, by Stokes’ Theorem, implies that

$$\begin{aligned} \hbox {vol}\,(\partial B(R))-\hbox {vol}\,(\partial B(1))\le \int _{B(R)\backslash B(1)} m(r). \end{aligned}$$

Let $\varepsilon >0$. Then we can find an $R_\varepsilon$ such that $m(r)<\varepsilon$ for $r>R_\varepsilon$. Setting $f(R)=V(R)$, we obtain

$$\begin{aligned} f'(R)\le \hbox {vol}\,(\partial B(1))+\int _{B(R_\varepsilon )\backslash B(1)} m(r) +\varepsilon (f(R)-f(R_\varepsilon )) \end{aligned}$$

for any $R>R_\varepsilon$. Thus

$$\begin{aligned} (e^{-\varepsilon R}(f(R)-f(R_\varepsilon )))'\le C_\varepsilon e^{-\varepsilon R} \end{aligned}$$

for $R>R_\varepsilon$, where $C_\varepsilon$ is a constant depending on $\varepsilon$ and the manifold $M$. Integrating from $R_\varepsilon$ to $R$, we obtain

$$\begin{aligned} f(R)< f(R_\varepsilon )+C_\varepsilon \varepsilon ^{-1}e^{-\varepsilon R_\varepsilon }e^{\varepsilon R} \end{aligned}$$

for $R>R_\varepsilon$. Thus for any $R$, we have

$$\begin{aligned} V(R)=f(R)<C(\varepsilon ) e^{\varepsilon R} \end{aligned}$$

for

$$\begin{aligned} C(\varepsilon )=f(R_\varepsilon )+C_\varepsilon \varepsilon ^{-1}e^{-\varepsilon R_\varepsilon }. \end{aligned}$$

$\square$

In other words, whenever the Laplacian of the radial function $r(x)=d(x,p)$ is asymptotically nonnegative in the sense of distribution, the manifold has subexponential volume growth with respect to the point $p$. In the case of finite volume for the manifold $M$, we will also need an assumption on the decay rate of the volume of a ball of radius $r$. We say that the volume of $M$ decays exponentially at $p$, if there exists an $\varepsilon _o>0$ such that

$$\begin{aligned} \mathrm{vol}\,(M)-V(r)\le e^{-\varepsilon _o r} \end{aligned}$$

for all $r$ large. For the purpose of computing the $L^2$ essential spectrum, we will need that the volume does not decay exponentially.

5.2 $L^1$ estimates for $\Delta \tilde{r}$

We set $\tilde{r}$ to be the smoothing of $r$ from Proposition 4.1. The following lemma is a more precise version of [16, Lemma 2].

Lemma 5.4

Let $r(x)$ be the radial function to a fixed point $p$ on $M$, and suppose that (8) is valid in the sense of distribution. Then we have the following two cases

• (a) Whenever $\mathrm{vol}\,(M)$ is infinite, for any $\varepsilon >0$ and $r_1>0$ large enough, there exists a $K=K(\varepsilon , r_1)$ such that for any $r_2>K$, we have

  $$\begin{aligned} \int _{B(r_2)\setminus B(r_1)} |\Delta \tilde{r}| \le \varepsilon \, V(r_2+1) ; \end{aligned}$$

  (10)

• (b) Whenever $\mathrm{vol}\,(M)$ is finite, for any $\varepsilon >0$ there exists a $K(\varepsilon )>0$ such that for any $r_2>K$, we have

  $$\begin{aligned} \int _{M \setminus B(r_2)} |\Delta \tilde{r}| \le \varepsilon \, (\mathrm{vol}\,(M)-V(r_2))+2\mathrm{vol}\,(\partial B(r_2)). \end{aligned}$$

Proof

By Proposition 4.1 and using the idea in the proof of Lemma 5.1, we obtain

$$\begin{aligned} |\Delta \tilde{r}(x)|\le 2\left( \max _{y\in B_x(1)} \Delta r(y)+\eta (r(x))+|b(x)|\right) -\Delta \tilde{r}(x) \end{aligned}$$

in the sense of distribution. Using our assumptions on $\Delta r$ and $\eta$, we see that for any $\varepsilon >0$ we can find an $r_1>0$ large enough such that whenever $r(x)>r_1$, then

$$\begin{aligned} 2\left( \max _{y\in B_x(1)} \Delta r(y)+\eta (r(x))\,\right) <\varepsilon /2 \end{aligned}$$

also in the sense of distribution. Therefore for $r>r_1+2$,

$$\begin{aligned} \int _{B(r)\setminus B(r_1)} |\Delta \tilde{r}| \le \frac{\varepsilon }{2}\, (V(r)-V(r_1))+2\int _{M\setminus B(r_1)}|b|-\int _{B(r)\setminus B(r_1)}\Delta \tilde{r}. \end{aligned}$$

Using Stokes’ Theorem, we get

$$\begin{aligned} \int _{B(r)\setminus B(r_1)}|\Delta \tilde{r}|\le \frac{\varepsilon }{2}\, (V(r)-V(r_1))+2\int _{M{\setminus B(r_1)}} |b|-\int _{\partial B(r)}\frac{\partial \tilde{r}}{\partial n} + \int _{\partial B(r_1)}\frac{\partial \tilde{r}}{\partial n}, \end{aligned}$$

where $\partial /\partial n$ is the outward normal direction on the boundary. Obviously, the above implies that

$$\begin{aligned} \int _{B(r)\setminus B(r_1)}|\Delta \tilde{r}|&\le \frac{\varepsilon }{2}\, (V(r)-V(r_1))+2\int _{M{\setminus B(r_1)}}|b| \nonumber \\&+\int _{\partial B(r)}\left| \frac{\partial \tilde{r}}{\partial n} -1\right| + \int _{\partial B(r_1)}\frac{\partial \tilde{r}}{\partial n}. \end{aligned}$$

(11)

We first consider the case when the volume of $M$ is infinite. By Proposition 4.1, choosing $r_1$ large enough we obtain

$$\begin{aligned} \int _{M{\setminus B(r_1)}}|b|< \frac{\varepsilon }{4} \end{aligned}$$

and

$$\begin{aligned} \Vert \nabla \tilde{r}-\nabla r\Vert _{L^1(M\backslash B(r_1))}\le 1 \end{aligned}$$

(12)

Since the volume of $M$ is infinite, then there exists $K=K(\varepsilon , r_1)>r_1+2$ such that whenever $r>K$

$$\begin{aligned} \int _{B(r)\setminus B(r_1)}|\Delta \tilde{r}|\le \frac{3\varepsilon }{4}\,( V(r){-V(r_1)\,})+\int _{\partial B(r)}\left| \frac{\partial \tilde{r}}{\partial n}-1\right| . \end{aligned}$$

(13)

We choose an $r'$ such that $|r'-r|<\frac{1}{2}$ and

$$\begin{aligned} \int _{\partial B(r')}\left| \frac{\partial \tilde{r}}{\partial n}-1\right| \le \int _{r-\frac{1}{2}}^{r+ \frac{1}{2}}\int _{\partial B(t)}\left| \frac{\partial \tilde{r}}{\partial n}-1\right| dt. \end{aligned}$$

By (12), we have

$$\begin{aligned} \int _{\partial B(r')}\left| \frac{\partial \tilde{r}}{\partial n}-1\right| <2. \end{aligned}$$

Therefore,

$$\begin{aligned} \int _{B(r')\setminus B(r_1)}|\Delta \tilde{r}|\le \frac{3\varepsilon }{4}\, (V(r')-V(r_1))+2. \end{aligned}$$

Choosing a possibly larger $K(\varepsilon ,r_1)$ we get (a).

The proof of (b) is similar. We choose $\eta (r)$ decreasing to zero so fast so that

$$\begin{aligned} \int _{M\backslash B(r_1)}|b|\le \frac{\varepsilon }{8}(\mathrm{vol}\,(M)- V(r_1)). \end{aligned}$$

Since the volume of $M$ is finite, sending $r\rightarrow \infty$ in (11) we have

$$\begin{aligned} \int _{M\setminus B_p(r_1)}|\Delta \tilde{r}|\le \varepsilon \, (\mathrm{vol}\,(M)-V(r_1))+ \int _{\partial B_p(r_1)}\frac{\partial \tilde{r}}{\partial n}. \end{aligned}$$

Since $|{\partial \tilde{r}}/{\partial n}|\le 2$ by (c) of Proposition 4.1, the lemma follows. $\square$

Corollary 5.1.

Suppose that $(i), (ii), (iii)$ hold on $M$ as in Lemma 5.2. Then the same integral estimates for $\Delta \tilde{r}$ hold as in Lemma 5.4.

6 The $L^2$ spectrum

In this section,we let $\tilde{r}(x)$ be the smoothing function defined in Proposition 4.1 of the radial function $r(x)=d(x,p)$. For each $i\in {\mathbb {N}}$, let $x_i, y_i, R_i, \mu _i$ be large positive numbers such that $x_i>2R_i>2\mu _i+4$ and $y_i>x_i+2R_i$. We take the cut-off functions $\chi _i: {\mathbb {R}}^+\rightarrow {\mathbb {R}}^+$, smooth with support on $[x_i/R_i-1, y_i/R_i+1]$ and such that $\chi _i=1$ on $[x_i/R,y_i/R]$ and $|\chi '_i|, |\chi ''_i|$ bounded. Let $\lambda >0$ be a positive number. We let

$$\begin{aligned} \phi _i(x)=\chi _i({\tilde{r}}/{R_i})\, e^{\sqrt{-1}\sqrt{\lambda }\,\tilde{r}}. \end{aligned}$$

(14)

Setting $\phi =\phi _i,\,R=R_i,\,x=x_i$ and $\chi =\chi _i$, we compute

$$\begin{aligned} \Delta \phi +\lambda \phi =&\left( R^{-2}\chi ''(\tilde{r}/R)+2i\sqrt{\lambda }R^{-1}\chi '(\tilde{r}/R)\right) e^{\sqrt{-1}\sqrt{\lambda }\tilde{r}}|\nabla \tilde{r}|^2\\&-\lambda \phi (|\nabla \tilde{r}|^2-1)+(R^{-1}\chi '(\tilde{r}/R)+i\sqrt{\lambda }\chi )e^{\sqrt{-1}\sqrt{\lambda }\tilde{r}}\Delta \tilde{r}. \end{aligned}$$

Then we have

$$\begin{aligned} |\phi |\le 1,\quad |\Delta \phi +\lambda \phi |\le \frac{C}{R}+C|\Delta \tilde{r}| +C|\nabla \tilde{r}-\nabla r|, \end{aligned}$$

(15)

where $C$ is a constant depending only on $\lambda$ and $M$.

Denote the inner product on $L^2(M)$ by $( \cdot \,,\,\cdot )$. We have the following key estimates:

Lemma 6.1.

Suppose that (8) is valid for the radial function $r$ in the sense of distribution. In the case that the volume of $M$ is finite, we make the further assumption that its volume does not decay exponentially at $p$. Then there exist sequences of large numbers $x_i, y_i, R_i, \mu _i$ such that the supports of the $\phi _i$ are disjoint and

$$\begin{aligned} \frac{\Vert (\Delta +\lambda )\phi _i\Vert _{L^1}}{(\phi _i,\phi _i)}\rightarrow 0 \end{aligned}$$

as $i\rightarrow \infty$.

Proof

The proof is similar to that of [16]. We define $x_i,y_i,R_i,\mu _i$ inductively. If $(x_{i-1}, y_{i-1}, R_{i-1}, \mu _{i-1})$ are defined, then we only need to let $\mu _i$ large enough so that the support of $\phi _i$ is disjoint with the previous $\phi _j$’s. For simplicity we suppress the $i$ in our notation. The upper bound estimates for $|\phi |$ and $|\Delta \phi +\lambda \phi |$ given in (15) imply that

$$\begin{aligned} \int _M(\phi , \Delta \phi +\lambda \phi )&\le \frac{C}{R} \,[V(y+R) -V(x-R)] \nonumber \\&+\, C \int _{B(y+R)\setminus B(x-R)} |\Delta \tilde{r}| +\eta (x-R). \end{aligned}$$

(16)

When the volume of $M$ is infinite, we choose a function $\eta$ as in Proposition 4.1 such that $\eta \le 1$. By Lemma 5.4, if we choose $R, x$ large enough but fixed, then for any $y>0$ large enough we have

$$\begin{aligned} \int _M(\phi , \Delta \phi +\lambda \phi ) \le 2 \varepsilon \,V(y+R+1). \end{aligned}$$

Since $\Vert \phi \Vert _2^2\ge V(y)-V(x)$, if we choose $y$ large enough, $\Vert \phi \Vert _2^2\ge \frac{1}{2} V(y).$ The subexponential volume growth of $M$ at $p$ that was proved in Lemma 5.3 implies that there exists a sequence of $y_k\rightarrow \infty$ such that $V(y_k+R+1)\le 2\, V(y_k)$. If not, then for a fixed number $y$ and for all $k\in {\mathbb {N}}$ we have that

$$\begin{aligned} V(y+k(R+1))> 2^k\, V(y). \end{aligned}$$

However, by the subexponential volume growth of the manifold

$$\begin{aligned} 2^k\, V(y) < V(y+k(R+1))\le C(\varepsilon _1) \, e^{\varepsilon _1 y} \, e^{k \,\varepsilon _1 (R+1)} \end{aligned}$$

for any $\varepsilon _1>0$ and $k$ large. This leads to a contradiction when we choose $\varepsilon _1$ such that $\varepsilon _1 R <\log 2$. Therefore, there exists a $y$ such that

$$\begin{aligned} V(y+R +1)\le 2\,V(y)\le 4 \Vert \phi \Vert _2^2. \end{aligned}$$

Combing the above inequalities, we have

$$\begin{aligned} \int _M(\phi , \Delta \phi +\lambda \phi ) \le 8 \varepsilon \Vert \phi \Vert _2^2. \end{aligned}$$

We now consider the finite volume case. Using equation (16) and Lemma 5.4 we obtain for $x-R>K(\varepsilon )$

$$\begin{aligned} \int _M(\phi , \Delta \phi +\lambda \phi ) \le&(R^{-1} + \varepsilon ) \,[\mathrm{vol}\,(M) -V(x-R)]\\&+\, 2 C \,\mathrm{vol}\,(\partial B(x-R)) +\eta (x-R). \end{aligned}$$

We set $h(r)=\mathrm{vol}\,(M) -V(r)$, a decreasing function. We choose $\eta (r)$ as in Proposition 4.1 so that $\eta (r)\le \frac{\varepsilon }{8} h(r)$. Making $\varepsilon$ even smaller and choosing $R$ and $x-R$ large enough, we get

$$\begin{aligned} \int _M(\phi , \Delta \phi +\lambda \phi ) \le \varepsilon \,h(x-R) - 2 C \,h'(x-R). \end{aligned}$$

Given that $\Vert \phi \Vert _2^2\ge h(x)-h(y)$ and the volume of $M$ is finite, we can choose $y$ large enough so that

$$\begin{aligned} \Vert \phi \Vert _2^2\ge \frac{1}{2} h(x). \end{aligned}$$

We would like to prove in this case that there exists a sequence of $x_k\rightarrow \infty$ such that

$$\begin{aligned} \varepsilon \,h(x_k-R) - 2C\,h'(x_k-R)\le 2 \varepsilon h(x_k). \end{aligned}$$

If the above inequality does not hold, then for all $x$ large enough

$$\begin{aligned} \varepsilon \,h(x-R) -2C \,h'(x-R) > 2 \varepsilon h(x). \end{aligned}$$

Replacing $\varepsilon$ by $\varepsilon /2C$, we obtain

$$\begin{aligned} \varepsilon \,h(x-R) - \,h'(x-R) > 2 \varepsilon h(x). \end{aligned}$$

This implies that

$$\begin{aligned} - \bigl (e^{-\varepsilon x}h(x-R)\bigr )' > 2 \varepsilon h(x) \, e^{-\varepsilon x}. \end{aligned}$$

Integrating from $x$ to $x+R$ and using the monotonicity of $h$ we have

$$\begin{aligned} h(x-R) >2(1 -e^{-\varepsilon R}) h(x+R). \end{aligned}$$

Choosing $R$ even larger, we can make $2 {(1-e^{-\varepsilon R})}> 5/4$, therefore

$$\begin{aligned} h(x-R) > \frac{5}{4} h(x+R) \end{aligned}$$

for all $x$ large enough. By iterating this inequality, we get for all positive integers $k$

$$\begin{aligned} h(x-R) > \left( \frac{5}{4}\right) ^k\, h(x+(2k-1)R). \end{aligned}$$

Therefore

$$\begin{aligned} \mathrm{vol}\,(M)-V(x-R) > \left( \frac{5}{4}\right) ^k [\mathrm{vol}\,(M)-V(x+(2k-1)R)] \end{aligned}$$

which gives

$$\begin{aligned} \mathrm{vol}\,(M)-V(x+(2k-1)R) \le \left( \frac{4}{5}\right) ^k \, \mathrm{vol}\,(M). \end{aligned}$$

Sending $k\rightarrow \infty$ this contradicts the nonexponential decay assumption on the volume. $\square$

Corollary 5.1 gives

Corollary 6.1

Suppose that (i), (ii), (iii) hold on $M$ as in Lemma 5.2. In the case that the volume of $M$ is finite, we make the further assumption that its volume does not decay exponentially at $p$. Then there exist sequences of large numbers $x_i, y_i, R_i, \mu _i$ and cut-off functions $\chi _i$ such that the supports of the $\phi _i$ are disjoint and

$$\begin{aligned} \frac{\Vert (\Delta +\lambda )\phi _i\Vert _{L^1}}{(\phi _i,\phi _i)}\rightarrow 0 \end{aligned}$$

as $i\rightarrow \infty$.

Now we prove Theorem 1.3. In fact we will be able to prove a more general, albeit more technical result:

Theorem 6.2

Let $M$ be a complete noncompact Riemannian manifold. Suppose that, with respect to a fixed point $p$, the radial function $r(x)=d(x,p)$ satisfies

$$\begin{aligned} \overline{\lim _{r\rightarrow \infty }}\; \Delta r\le 0 \end{aligned}$$

in the sense of distribution, and if the volume of the manifold is finite, we additionally assume that its volume does not decay exponentially at $p$. Then the $L^2$ spectrum of the Laplace operator on functions is $[0,\infty )$.

Proof

Let $\phi _i$ be the sequence of functions as defined in  (14). Then by the construction of the functions and Corollary 6.1, the assumptions of Theorem 1.1 are satisfied. This completes the proof of the theorem. $\square$

Remark 6.3

We note that a similar result should hold on warped product manifolds $M={\mathbb {R}}\times _J \tilde{M}$ with metric $g=d\rho ^2 + J^2(\rho ,\theta ) \, \tilde{g},$ where $(\tilde{M},\tilde{g})$ is a compact $(n-1)$-dimensional submanifold of $M$ and $\rho$ is the distance function from this submanifold. Under the same asymptotically nonnegative assumption on $\hbox {Ric}(\partial \rho , \partial \rho )$ as in Lemma 5.2, we also get that the $L^2$ spectrum of the Laplace operator on functions is $[0,\infty )$.

7 Complete shrinking Ricci solitons

A noncompact complete Riemannian manifold $M$ with metric $g$ is called a gradient shrinking Ricci soliton if there exists a smooth function $f$ such that the Ricci tensor of the metric $g$ is given by

$$\begin{aligned} R_{ij}+\nabla _i \nabla _j f = \rho \, g_{ij} \end{aligned}$$

for some positive constant $\rho >0$. By rescaling the metric we may rewrite the soliton equation as

$$\begin{aligned} R_{ij}+\nabla _i \nabla _j f = \frac{1}{2} \, g_{ij}. \end{aligned}$$

The scalar curvature $R$ of a gradient shrinking Ricci soliton is nonnegative, and the volume growth of such manifolds (with respect to the Riemannian metric) is Euclidean. Hamilton [12] proved that the scalar curvature of a gradient shrinking Ricci soliton satisfies the equations

$$\begin{aligned} \nabla _i R = 2 \, R_{ij}\, \nabla _j f,\\ R+|\nabla f|^2-f=C_o \end{aligned}$$

for some constant $C_o$. We may add a constant to $f$ so that

$$\begin{aligned} R+|\nabla f|^2-f=0. \end{aligned}$$

In [16], the authors proved that

• (1) the $L^1$ essential spectrum of the Laplacian contains $[0, \infty )$;

• (2) the $L^2$ essential spectrum of the Laplacian is $[0, \infty )$, if the scalar curvature has sub-quadratic growth.

Using our new Weyl Criterion, we are able to remove the curvature condition.

Proof of Theorem 1.2

It can be shown that $f(x)\ge 0$ and the key idea is to use $\rho (x)=2\sqrt{f(x)}$ as an approximate distance function on the manifold, because of the special properties that it satisfies.

We define

$$\begin{aligned} D(r)=\{x\in M : \rho (x)<r\} \end{aligned}$$

and set $V(r) =\mathrm{vol}\,(D(r))$. For some positive number $y$ sufficiently large we consider the cut-off function $\chi : {\mathbb {R}}^+\rightarrow {\mathbb {R}}$, smooth with support in $[0, y +2]$ and such that $\chi =1$ on $[1,y+1]$ and $|\chi '|, |\chi ''|\le C$. For any $\lambda > 0$ and large enough constants $b, l$ we let

$$\begin{aligned} \phi (\rho )=\chi \left( \frac{\rho -b}{l}\right) \, e^{\sqrt{-1}\sqrt{\lambda }\,\rho } \end{aligned}$$

which has support on $[b+l, b+l(y+1)]$. Lu and Zhou [16, page 3289] demonstrate that for sufficiently large $l$ and $b$

$$\begin{aligned} \int _M |\Delta \phi +\lambda \phi | \le \varepsilon V(b+(y+2)l). \end{aligned}$$

At the same time

$$\begin{aligned} \Vert \phi \Vert _{L^2}^2\ge V(b+(y+1)l)-V(b+l) \end{aligned}$$

(note that the same holds true for the $L^1$ norm of $\phi$). Arguing as in [16, Theorem 6] we conclude that there exists a $y$ large enough such that

$$\begin{aligned} \int _M |\Delta \phi +\lambda \phi | \le 4 \varepsilon \Vert \phi \Vert _{L^2}^2. \end{aligned}$$

As in the previous section, we may also choose appropriate sequences of $b_i, l_i$ such that the supports of the $\psi _i$ are disjoint and condition (2) of Theorem 1.1 holds. Condition (1) is verified by the estimate above and the fact that $\Vert \phi _i\Vert _{L^\infty }=1$. $\square$

8 Exhaustion functions on complete manifolds

From what we have seen so far, it is apparent that two things are important when computing the essential spectrum of the Laplacian:

• (1) The control of the $L^1$ norm of $\Delta r$;

• (2) The control of the volume growth and decay of geodesic balls.

The same idea can be used for manifolds whose essential spectrum is not the half real line.

In the spirit of the results above, we are also able to modify a theorem of Elworthy and Wang [8]. We now consider manifolds on which there exists a continuous exhaustion function $\gamma \in {\mathcal {C}}(M)$ such that

• (a) $\gamma$ is unbounded above and is ${\mathcal {C}}^2$ smooth in the domain $\{\gamma >R\}$ for some $R>0$ and

• (b) $\mathrm{vol}\,(\{m_o<\gamma <n\})<\infty$ for some $m_o$ and any $n>m_o$ where the volume is measured with respect to the Riemannian metric.

For $t>0$ and $c\in {\mathbb {R}}$ we define $B_t=\{\gamma (x)<t\}$ and set $dv_c=e^{-c\gamma }dv$. For $t\ge s$, let $U_c(s,t)=\mathrm{vol}\,_c(B_t{\setminus } B_s)$ where $\mathrm{vol}\,_c$ is the volume with respect to the measure $dv_c$.

We begin by stating the result of Elworthy and Wang for the sake of comparison.

Theorem 8.1

([8, Theorem 1.1]) Suppose that there exists a function $\gamma \in {\mathcal {C}}(M)$ that satisfies $(a)$ and $(b)$ and a constant $c\in {\mathbb {R}}$ such that

$$\begin{aligned} \lim _{s\rightarrow \infty } \overline{\lim _{t\rightarrow \infty }}\; U_c(s,t)^{-1}\int _{B_t\setminus B_s} [(\Delta \gamma -c)^2+(|\nabla \gamma |^2-1)^2 ] \, dv_c=0 \end{aligned}$$

(17)

and

$$\begin{aligned} \lim _{t\rightarrow \infty } \max \{U_c(m_o,t), U_c(t,\infty )^{-1} \} \,e^{-\varepsilon t}=0 \qquad \mathrm {for \ any\ } \varepsilon >0. \end{aligned}$$

(18)

Then $\sigma (-\Delta )\supset [c^2/4,\infty ).$ When the above hold for $c=0$, then $\sigma (-\Delta )= [0,\infty ).$

Note that condition (18) implies that when $c = 0$ the volume of the manifold grows and decays subexponentially, as was the case for us in the previous sections. The assumption here is that the weighted volume grows and decays subexponentially.

Our result is as follows:

Theorem 8.2.

Suppose that there exists a function $\gamma \in {\mathcal {C}}(M)$ that satisfies $(a)$ and $(b)$ and a constant $c\in {\mathbb {R}}$ such that

$$\begin{aligned} \lim _{s\rightarrow \infty } \overline{\lim _{t\rightarrow \infty }}\; \,U_{c}(s,t)^{-1} \,\int _{B_t\setminus B_s} (|\Delta \gamma -c|+|\,|\nabla \gamma |^2-1|) \, dv_{c}=0 \end{aligned}$$

(19)

and

$$\begin{aligned} \lim _{t\rightarrow \infty } \max \{U_c(m_o,t), U_c(t,\infty )^{-1} \} \,e^{-\varepsilon t}=0 \qquad \mathrm {for \ any\ } \varepsilon >0. \end{aligned}$$

(20)

If (19) and (20) hold for $c=0$, then $\sigma (-\Delta )=[0,\infty ).$

In the case they hold for $c\ne 0$, we make the additional assumptions that the heat kernel of the Laplacian satisfies the pointwise bound

$$\begin{aligned} p_t(x,y)\le C t^{-m}\,e^{-\frac{(\gamma (x)-\gamma (y))^2}{4C_1 t} -\frac{d(x,y)^2}{4C_2 t } +\beta _1|\gamma (x)-\gamma (y)|+\beta _2d(x,y)+\beta _3 t} \end{aligned}$$

(21)

for some positive constants $m, C_1, C_2, \beta _1, \beta _2, \beta _3$, and that the Ricci curvature of the manifold is bounded below $\hbox {Ric}(M)\ge -(n-1)K$ for a nonnegative number $K$. Then $\sigma (-\Delta ) \supset [c^2/4,\infty ).$

In the case $c=0$, the main difference between our result and Theorem 1.1 of [8] is that we only need to control the $L^1$ norms of $|\Delta \gamma -c|$ and $| |\nabla \gamma |^2-1|$ as in (19), instead of their $L^2$ norms (compare to 17). Our assumption is weaker in various cases, for example when $\gamma$ is the radial function where we know that its Laplacian is not locally $L^2$ integrable whenever the manifold has a cut-locus, but it is locally $L^1$ integrable.

In the case $c\ne 0$, the additional assumption (21) is similar to requiring a uniform Gaussian bound for the heat kernel, but now with respect to the $\gamma$ function as well. Such a bound is certainly true in the case of hyperbolic space with $\gamma$ the radial function.

The proof uses similar estimates to those of Elworthy and Wang for the measures of annuli along the exhaustion function $\gamma$. We provide an outline of the argument with the necessary modifications.

Proof

Set $\lambda \ge c^2/4$ be a fixed number. For any $t>s$ we let $\chi : {\mathbb {R}}^+\rightarrow {\mathbb {R}}^+$, be a smooth cut-off function with support on $[s-1, t+1]$ and such that $\chi =1$ on $[s,t]$ and $|\chi '|, |\chi ''|$ bounded. Let $\lambda _c=\sqrt{\lambda -c^2/4}$ and define for $s\ge 0$

$$\begin{aligned} f(s)= e^{(i{\lambda _c} - c/2)\,s}. \end{aligned}$$

Consider the test function

$$\begin{aligned} \phi _{s,t}(x)=\chi (\gamma (x))\, f(\gamma (x)). \end{aligned}$$

We compute

$$\begin{aligned} \Delta \phi _{s,t} +\lambda \phi _{s,t} = (\chi ''f+2\chi 'f'+\chi f'')|\nabla \gamma |^2 + (\chi ' f + \chi f')\Delta \gamma +\lambda \,\chi f. \end{aligned}$$

Using the fact that $f'' +c f' +\lambda f=0$ we obtain

$$\begin{aligned} \Delta \phi _{s,t} +\lambda \phi _{s,t}&= (\chi ''f+2\chi 'f')|\nabla \gamma |^2 + (\chi ' f )\Delta \gamma + \chi f'(\Delta \gamma -c |\nabla \gamma |^2)\\&+\, \lambda \,\chi f(1-|\nabla \gamma |^2). \end{aligned}$$

Therefore there exists a constant $C$ such that

$$\begin{aligned} |\Delta \phi _{s,t} +\lambda \phi _{s,t}|\le Ce^{-c/2\gamma }\, {\bigl [}(|\Delta \gamma -c|+|\,|\nabla \gamma |^2-1|) 1_{\hbox {spt}(B_{{t+1}}\setminus B_{{s-1}})} +1_{\hbox {spt}(\chi ')} {\bigr ]}.\nonumber \\ \end{aligned}$$

(22)

For the rest of the estimates, we will repeatedly use

$$\begin{aligned} \lim _{s,t\rightarrow \infty } (U_c(s-1,s)+U_c(t,t+1))/U_c(s,t)=0, \end{aligned}$$

(23)

which follows from  (20).

Using (22), we have

$$\begin{aligned} |(\phi _{s,t},\Delta \phi _{s,t} +\lambda \phi _{s,t})|&\le C \int _{B_{{t+1}}\backslash B_{{s-1}}}(|\Delta \gamma -c|+|\,|\nabla \gamma |^2-1|)\,dv_c \nonumber \\&{+\, C (U_c(s-1,s)+U_c(t,t+1))}. \end{aligned}$$

(24)

We observe that

$$\begin{aligned}&\frac{1}{U_c(s,t)}\int _{B_{t+1}\backslash B_{s-1}}(|\Delta \gamma -c|+|\,|\nabla \gamma |^2-1|)\,dv_c \nonumber \\&\quad =\left[ 1+\frac{U_c(s-1,s)+U_c(t,t+1)}{U_c(s,t)}\right] \nonumber \\&\qquad \cdot \,\frac{1}{U_c(s-1,t+1)}\int _{B_{t+1}\backslash B_{s-1}}(|\Delta \gamma -c|+|\,|\nabla \gamma |^2-1|)\,dv_c, \end{aligned}$$

which tends to zero as $s,t\rightarrow \infty$ by (23) and assumption (19). Since $\Vert \phi _{s,t}\Vert ^2_{L^2}\ge U_c(s,t)$, inequality (24), the above estimate and  (23) imply that

$$\begin{aligned} \lim _{s,t\rightarrow \infty } |(\phi _{s,t}, \Delta \phi _{s,t} +\lambda \phi _{s,t})|/\Vert \phi _{s,t}\Vert ^2_{L^2}=0. \end{aligned}$$

(25)

When $c=0$, we choose appropriate sequences of $s_n, t_n \rightarrow \infty$ such that condition (2) of Theorem 1.1 holds. Condition (1) of the Corollary follows from (25) and the fact that the functions $\phi _{s_n,t_n}$ are bounded. Therefore, $\lambda _0=\sqrt{\lambda }$ belongs to the essential $L^2$ spectrum. Given that $\lambda$ is any nonnegative number, the result follows.

In the case $c\ne 0$, we will apply Corollary 2.2. For a fixed natural number $i>m$ and $\alpha >0$ we have that the integral kernel of $(-\Delta +\alpha )^{-i},\,g_{\alpha }^{i}(x,y)$, is given by

$$\begin{aligned} g_{\alpha }^{i}(x,y)= C(n)\int _0^\infty p_t(x,y)\, t^{i-1} \, e^{-\alpha t} \, dt. \end{aligned}$$

On the other hand, it is a property of the exponential function that for any $\beta _4, \beta _5 \in {\mathbb {R}}$

$$\begin{aligned} e^{-\frac{(\gamma (x)-\gamma (y))^2}{4C_1 t}} \le e^{-\beta _4|\gamma (x)-\gamma (y)|} e^{C_1\,\beta _4^2 t} \end{aligned}$$

and

$$\begin{aligned} e^{-\frac{d(x,y)^2}{4C_2 t}} \le e^{-\beta _5 d(x,y)} e^{C_2\,\beta _5^2 t}. \end{aligned}$$

Combining the above, we have that for any $N>m$ and $\beta _4, \beta _5>0$ there exists an $\alpha >0$ large enough, and a constant $C$ such that

$$\begin{aligned} g_{\alpha }^{i}(x,y)\le C\,e^{-\beta _4 \, |\gamma (x)-\gamma (y)|-\beta _5 d(x,y)} \end{aligned}$$

for $i=N, N+1$. As a result, for any $t>s>2$

$$\begin{aligned} \int _{B_{t+1}\setminus B_{s-1}} g_{\alpha }^{i}(x,y) e^{-c/2\gamma (y)} dy&\le C \int _{B_{t+1}\setminus B_{s-1}} e^{-\beta _4 |\gamma (x)-\gamma (y)| -\beta _5 d(x,y)} \, e^{-c/2\gamma (y)} dy \\&\le C\,e^{-c/2\gamma (x)} \end{aligned}$$

after choosing $\beta _4=|c|/2$ and $\beta _5>\sqrt{K}$. This estimate together with (23) also give

$$\begin{aligned} |((-\Delta +\alpha )^{-i}\phi _{s,t},\Delta \phi _{s,t} +\lambda \phi _{s,t})|&\le C \int _{B_{{t+1}}\backslash B_{{s-1}}}(|\Delta \gamma -c|+|\,|\nabla \gamma |^2-1|)\,dv_c \\&+\, C (U_c(s-1,s)+U_c(t,t+1)). \end{aligned}$$

As a result,

$$\begin{aligned} \lim _{s,t\rightarrow \infty } |((-\Delta +\alpha )^{-i}\phi _{s,t},\Delta \phi _{s,t} +\lambda \phi _{s,t})|/\Vert \phi _{s,t}\Vert ^2_{L^2}=0. \end{aligned}$$

(26)

Choosing appropriate sequences of $s_n, t_n \rightarrow \infty$ and setting $\psi _n = \phi _{s_n,t_n}/\Vert \phi _{s_n,t_n}\Vert _{L^2}$, conditions (1) and (4) of Corollary 2.2 hold for the functions $\psi _n$. That (2) and (3) also hold follows from (25) and (26) respectively. $\square$

9 The use of continuous test functions

In this section we will see that it is not necessary to use cut-off functions in our test functions. We will do that by first proving yet another version of the generalized Weyl’s criterion (Corollary 9.1). This version of Weyl’s Criterion sometimes provides a cleaner method for computing the essential spectrum.

Let $D$ be a bounded domain of $M$ with smooth boundary. We use the notation ${\mathcal {C}}_0^\infty (D)$ to denote the set of smooth functions on the closure $\bar{D}$ which vanish on the boundary $\partial D$. Let $\rho :D\rightarrow {\mathbb {R}}$ be the distance function to the boundary $\partial D$.

Definition 9.1

We define ${\mathcal {C}}_0^+( D)$ to be the set of functions $f$ on $D$ with the properties

• (1) $f$ is continuous, vanishing on $\partial D$;

• (2) $f$ is Lipschitz, $\nabla f$ is essentially bounded, and $|\Delta f|$ exists in the sense of distribution;

• (3) As $\varepsilon \rightarrow 0,\,\int _{\{\rho \le \varepsilon \}}|f|\le \frac{1}{2}\varepsilon ^2(\int _{\partial D} |\nabla f|+o(1))$, and $\int _{\{\rho \le \varepsilon \}}|\nabla f|\le \varepsilon (\int _{\partial D} |\nabla f|+o(1))$.

Let ${\mathcal {C}}_0^+(M)$ be the set of continuous functions whose support is a bounded domain of $M$ with smooth boundary and

$$\begin{aligned} f\in {\mathcal {C}}_0^+(\hbox {supp}f). \end{aligned}$$

We have the following:

Corollary 9.1

A nonnegative real number $\lambda$ belongs to the spectrum $\sigma (-\Delta )$, if there exists a sequence $\{\psi _n\}_{n \in {\mathbb {N}}}$ of functions in ${\mathcal {C}}_0^+(M)$ such that

• (1) $\frac{\Vert \psi _n\Vert _{L^\infty (D_n)}\cdot \,\,(\Vert (-\Delta -\lambda )\psi _n\Vert _{L^1( D_n)}+\Vert \nabla \psi _n\Vert _{L^1(\partial D_n)})}{\Vert \psi _n\Vert _{L^2(D_n)}^2} \rightarrow 0, \hbox { as } n\rightarrow \infty ,$

where $D_n= \hbox {supp}\,{\psi _n}$. Moreover, $\lambda$ belongs to $\sigma _\mathrm{ess}(-\Delta )$ of $\Delta$, if

• (2) For any compact subset $K$ of $M$, there exists an $n$ such that the support of $\psi _n$ is outside $K$.

The above corollary can be proved using the following approximation result:

Proposition 9.1.

Let $f\in {\mathcal {C}}_0^+(M)$. Then for any $\varepsilon >0$, there exists a smooth function $h$ of $M$ such that

• (a) $\hbox {supp}\,(h)\subset \hbox {supp}\, (f)$;

• (b) $\Vert f-h\Vert _{L^1}+\Vert f-h\Vert _{L^2}\le \varepsilon$;

• (c) $\Vert (-\Delta -\lambda ) h\Vert _{L^1}\le C (\Vert (-\Delta -\lambda ) f\Vert _{L^1( D)}+\Vert \nabla f\Vert _{L^1(\partial D)})$,

where $C$ is a constant independent of $f$, and $D=\hbox {supp}\,(f)$.

Proof

Let $\chi (t)$ be a cut-off function that vanishes in a neighborhood of 0 and is 1 for $t\ge 1$. Let $\delta >0$ be a small number. Consider

$$\begin{aligned} g_\delta (x)=\chi \left( \frac{\rho (x)}{\delta }\right) f(x). \end{aligned}$$

It is not difficult to prove (a), (b) in the Proposition when we replace $h$ by $g_\delta$. To prove (c) we compute

$$\begin{aligned} (-\Delta -\lambda ) g_\delta =\chi (-\Delta -\lambda ) f-2\delta ^{-1}\chi '\nabla \rho \nabla f -(\delta ^{-2}\chi ''+\delta ^{-1}\chi '\Delta \rho ) f. \end{aligned}$$

Since $\partial D$ is smooth, $\rho$ is a smooth function near $\partial D$. Therefore by (3) of Definition 9.1 we have

$$\begin{aligned} \Vert (-\Delta -\lambda ) g_\delta \Vert _{L^1}\le C (\Vert (-\Delta -\lambda ) f\Vert _{L^1( D)}+\Vert \nabla f\Vert _{L^1(\partial D)}) \end{aligned}$$

for $\delta$ sufficiently small.

The proof that $g_\delta$ can be approximated by a smooth function is similar to that of Proposition 4.1. We sketch the proof here.

Let $D=\cup U_i$ be a finite cover of $D$. Without loss of generality, we assume that those $U_i$’s which intersect with $\partial D$ are outside the support of $g_\delta$. Let $\mathbf{x_i}=(x_i^1,\ldots ,x_i^n)$ be the local coordinates of $U_i$. Define $g_i=g_\delta |_{U_i}$.

Let $\xi (\mathbf x)$ be a non-negative smooth function of ${\mathbb {R}}^n$ whose support is within the unit ball. Assume that

$$\begin{aligned} \int _{{\mathbb {R}}^n}\xi =1. \end{aligned}$$

Without loss of generality, we assume that each $U_i$ is an open subset of the unit ball of ${\mathbb {R}}^n$ with coordinates $\mathbf{x_i}$. Then for any $\varepsilon >0$,

$$\begin{aligned} g_{i,\varepsilon }=\frac{1}{\varepsilon ^n}\int _{{\mathbb {R}}^n}\xi \left( \frac{\mathbf{x_i}-\mathbf{y_i}}{\varepsilon }\right) g_i(\mathbf{y_i}) d\mathbf{y_i} \end{aligned}$$

is a smooth function on $U_i$ and hence on $M$. Let $\{\sigma _i\}$ be a sequence of positive numbers such that

$$\begin{aligned} \sum _i\sigma _i(|\Delta \psi _i(x)|+4|\nabla \psi _i (x)|+\psi _i(x)) \end{aligned}$$

(27)

is sufficiently small. By [11, Lemma 7.1, 7.2], for each $i$, we can choose $\varepsilon _i<1$ small enough so that

$$\begin{aligned} \begin{aligned}&|g_{i,\varepsilon _i}(x)-g_i(x)|\le \sigma _i;\\&\Vert \nabla g_{i,\varepsilon _i}-\nabla g_i\Vert _{L^1{(U_i)}}\le \sigma _i. \end{aligned} \end{aligned}$$

(28)

We also have

$$\begin{aligned} \Vert \Delta g_{i,\varepsilon _i}\Vert _{L^1}\le \Vert \Delta g_i\Vert _{L^1}. \end{aligned}$$

(29)

Define

$$\begin{aligned} h=\sum _i\psi _i g_{i,\varepsilon _i}, \quad b=2 { \sum _i} \nabla \psi _i\cdot \nabla g_{i,\varepsilon _i}. \end{aligned}$$

Since $\sum _i (\nabla \psi _i\cdot \nabla g_i)=(\sum _i\nabla \psi _i) \cdot \nabla g_\delta =0$ almost everywhere on $D$, we have

$$\begin{aligned} b=2\sum _i \nabla \psi _i\cdot (\nabla g_{i,\varepsilon _i}-\nabla g_i). \end{aligned}$$

We compute

$$\begin{aligned} \Delta h=\sum _i[(\Delta \psi _i)\, g_{i,\varepsilon _i}+2\nabla \psi _i\nabla g_{i,\varepsilon _i}+\psi _i\Delta g_{i,\varepsilon _i}], \end{aligned}$$

and since

$$\begin{aligned} \sum _i(\Delta \psi _i ) g_i=\sum _i (\Delta \psi _i) g_\delta =0, \end{aligned}$$

we have

$$\begin{aligned} \Delta h=\sum _i[\Delta \psi _i(g_{i,\varepsilon _i}- g_i)+ 2\sum _i\nabla \psi _i\cdot (\nabla g_{i,\varepsilon _i}-\nabla g_i)+\psi _i\Delta g_{i,\varepsilon _i}]. \end{aligned}$$

By (28), (29), we may choose $\varepsilon _i$ to be sufficiently small so that

$$\begin{aligned} \Vert {(-\Delta -\lambda )} h\Vert _{L^1{(D)}}\le 2 \Vert (-\Delta -\lambda ) g_\delta \Vert _{L^1(D)}. \end{aligned}$$

$\square$