Eigenvalues of the Laplacian on a compact manifold with density
Abstract.
In this paper, we study the spectrum of the weighted Laplacian (also called BakryEmery or Witten Laplacian) on a compact, connected, smooth Riemannian manifold endowed with a measure . First, we obtain upper bounds for the th eigenvalue of which are consistent with the power of in Weyl’s formula. These bounds depend on integral norms of the density , and in the second part of the article, we give examples showing that this dependence is, in some sense, sharp. As a corollary, we get bounds for the eigenvalues of Laplace type operators, such as the Schrödinger operator or the Hodge Laplacian on forms. In the special case of the weighted Laplacian on the sphere, we get a sharp inequality for the first nonzero eigenvalue which extends Hersch’s inequality.
Key words and phrases:
Manifold with density, Weighted Laplacian, Witten Laplacian, Eigenvalue, Upper bound2010 Mathematics Subject Classification:
58J50, 35P15, 47A751. Introduction
In this article, our main aim is to study the spectrum of the weighted Laplacian (also called BakryEmery Laplacian) on a compact, connected, smooth Riemannian manifold endowed with a measure , where is a positive density and is the Riemannian measure induced by the metric . Such a triple is known in literature as a weighted Riemannian manifold, a manifold with density, a smooth metric measure space or a BakryEmery manifold. Denoting by and the gradient and the Laplacian with respect to the metric , the operator is defined by
so that, for any function , satisfying Neumann boundary conditions if ,
Note that is selfadjoint as an operator on and is unitarily equivalent (through the transform ) to the Schrödinger operator . That is why itself is sometimes called Witten Laplacian. , which is nothing but the restriction to functions of the Witten Laplacian associated to
Weighted manifolds arise naturally in several situations in the context of geometric analysis and their study has been very active in recent years. Their BakryEmery curvature plays a role which is similar in many respects to that played by the Ricci curvature for Riemannian manifolds, and appears as a centerpiece in the analysis of singularities of the Ricci flow in Perelman’s work (see [35, 36]). The weighted Laplacian appears naturally in the study of diffusion processes (see e.g., the pioneering work of Bakry and Emery [2]). Eigenvalues of are strongly related to asymptotic properties of mmspaces, such as the study of Levy families (see [11, 19, 34]). Without being exhaustive, we refer to the following articles and the references therein: [29, 30, 31, 37, 38, 39, 43] and, closely related to our topic, [1, 12, 13, 21, 32, 33, 41, 44, 45]
The spectrum of , with Neumann boundary conditions if , consists of an unbounded sequence of eigenvalues
which satisfies the Weyl’s asymptotic formula
where is the Riemannian volume of and is the volume of the unit ball in . The first aim of this paper is to obtain bounds for which are consistent with the power of in Weyl’s formula.
Before stating our results, let us recall some known facts about the eigenvalues of the usual Laplacian (case ). Firstly, the wellknown Hersch’s isoperimetric inequality (see [24]) asserts that on the dimensional sphere , the first positive normalized eigenvalue is maximal when is a “round” metric (see [9, 10, 25, 27, 40] for similar results on other surfaces). Korevaar [26] proved that on any compact manifold of dimension , is bounded above independently of . More precisely, if is a compact orientable surface of genus , then
(1) 
where is an absolute constant (see [20] for an improved version of this inequality). On the other hand, on any compact manifold of dimension , the normalized first positive eigenvalue can be made arbitrarily large when runs over the set of all Riemannian metrics on (see [4, 28]). However, the situation changes as soon as we restrict ourselves to a fixed conformal class of metrics. Indeed, on the sphere , round metrics maximize among all metrics which are conformally equivalent to the standard one (see [8, Proposition 3.1]). Furthermore, Korevaar [26] proved that for any compact Riemannian manifold one has
(2) 
where is a constant depending only on the conformal class of the metric . Korevaar’s approach has been revisited and placed in the context of metric measure spaces by Grigor’yan and Yau [16] and, then, by Grigor’yan, Netrusov and Yau [18].
The first observation we can make about possible extensions of these results to weighted Laplacians is that, given any compact Riemannian manifold , the eigenvalues cannot be bounded above independently of . Indeed, from the semiclassical analysis of the Witten Laplacian (see [22]), we can easily deduce that (Proposition 2.1) if is any smooth Morse function on with stable critical points, then the family of densities satisfies for ,
Therefore, any extension of the inequalities (1) and (2) to must necessarily have a density dependence in the righthand side. The following theorem gives such an extension in which the upper bound depends on the ratio between the norm and the norm of . In all the sequel, the norm of with respect to will be denoted by .
Theorem 1.1.
Let be a compact weighted Riemannian manifold. The eigenvalues of the operator , with Neumann boundary conditions if , satisfy :
(I) If , then, ,
where is a constant depending only on the conformal class of .
(II) if and is orientable of genus , then, ,
where is an absolute constant.
It is clear that taking in Theorem 1.1, we recover the inequalities (1) and (2). Moreover, as we will see in the next section, if is boundaryless and the conformal class contains a metric with nonnegative Ricci curvature, then , where is a constant which depends only on the dimension.
Notice that there already exist upper bounds for the eigenvalues of in the literature, but they usually depend on derivatives of , either directly or indirectly, through the BakryEmery curvature. The main feature of our result is that the upper bounds we obtain depend only on norms of the density. The proof of Theorem 1.1 relies in an essential way on the technique developed by Grigor’yan, Netrusov and Yau [18].
Regarding Hersch’s isoperimetric type inequalities, they extend to our context as follows (see Corollary 3.2): Given any metric on which is conformally equivalent to the standard metric , and any positive density , one has
where is the volume of the standard sphere and with the convention that when . Moreover, the equality holds in the inequality if and only if is constant and is a round metric.
Next, let us consider a Riemannian vector bundle over a Riemannian manifold and a Laplace type operator
acting on smooth sections of the bundle. Here is a connection on which is compatible with the Riemannian metric and is a symmetric bundle endomorphism (see e.g., [3, Section E]). The operator is selfadjoint and elliptic and we will list its eigenvalues as: Important examples of such operators are given by Schrödinger operators acting on functions (here is just the potential), the Hodge Laplacian acting on differential forms (in which case is the curvature term in Bochner’s formula), and the square of the Dirac operator ( being in this case a multiple of the scalar curvature). Another important example is the Witten Laplacian acting on differential forms, whose restriction to functions is precisely given by a weighted Laplacian, the main object of study of this paper.
In Section 4 we will prove (Theorem 4.1) an upper bound for the gap between the th eigenvalue and the first eigenvalue of involving integral norms of a first eigensection . For example, if , then
(3) 
where is a constant depending only on the conformal class of . The reason why we bound the gap instead of itself is due to the fact that, even when and is the standard Hodge Laplacian acting on forms, the first positive eigenvalue is not bounded on any conformal class of metrics (see [5]). For estimates on the gap when a finite group of isometries is acting, we refer to [7].
Inequality (3) should be regarded as an extension of Theorem 1.1. Indeed, if is the Schrödinger operator which is unitarily equivalent to the operator , then and any first eigenfunction of is a scalar multiple of . Thus, taking in (3) we recover the first estimate in Theorem 1.1.
Our main aim in section 5 is to discuss the accuracy of the upper bounds given in Theorem 1.1 regarding the way they depend on the density . That is why we exhibit an explicit family of compact manifolds , each endowed with a sequence of densities , and give a sharp lower estimate of the first positive eigenvalue in terms of . This enables us to see that both and the ratio tend to infinity linearly with respect to . Thus, we have
with as , and and are two positive constants which do not depend on .
The examples of densities we give are modeled on Gaussian densities (i.e. ) on . For example, if is a bounded convex domain in , we observe that for all . We then extend the lower bound to manifolds of revolution (at least asymptotically as ). However, in the case of a closed manifold of revolution, there is an additional difficulty coming from the fact that we need to extend smoothly this kind of density to the whole manifold in such a way as to preserve the estimates on both the eigenvalues and the norms.
2. Upper bounds for weighted eigenvalues in a smooth metric measure space and proof of Theorem 1.1
Let be a compact connected Riemannian manifold, possibly with a nonempty boundary. Let be a bounded nonnegative function on and let be a nonatomic Radon measure on with . To such a pair , we associate the sequence of nonnegative numbers given by
where is the set of all dimensional vector subspaces of and
In the case where is of class and , the variational characterization of eigenvalues of the weighted Laplacian gives (see e.g. [17])
(4) 
Theorem 1.1 is a direct consequence of the following
Theorem 2.1.
Let be a compact Riemannian manifold possibly with nonempty boundary. Let be a nonnegative function and let be a nonatomic Radon measure with .
(I) If , then for every , we have
where is a constant depending only on the conformal class of .
Moreover, if is closed and contains a metric with nonpositive Ricci curvature, then where is a constant depending only on .
(II) If is a compact orientable surface of genus , then for every , we have
where is an absolute constant.
The proof of this theorem is based on the method described by Grigor’yan, Netrusov and Yau in [18] and follows the same lines as the proof they have given in the case . The main step consists in the construction of a family of disjointly supported functions with controlled Rayleigh quotient.
Let us fix a reference metric and denote by the distance associated to . An annulus is a subset of of the form where and (if necessary, we will denote it ). The annulus is by definition the annulus .
To such an annulus we associate the function supported in and such that
We introduce the following constant:
where stands for the ball of radius centered at in . Notice that since is compact, the constant is finite and depends only on . This constant can be bounded from above in terms of a lower bound of the Ricci curvature and an upper bound of the diameter (BishopGromov inequality). In particular, if the Ricci curvature of is nonnegative, then is bounded above by a constant depending only on the dimension .
Lemma 2.1.
For every annulus one has
Proof.
Let be an annulus of . Since is supported in we get, using Hölder inequality,
From the conformal invariance of we have
with
Hence,
where the last inequality follows from the definition of . Putting together all the previous inequalities, we obtain the result of the Lemma. ∎
Proof of part (I) of Theorem 2.1: Let us introduce the constant , that we call the covering constant, defined to be the infimum of the set of all integers such that, for all , any ball of radius in can be covered by balls of radius . Again, the compactness of ensures that is finite, and BishopGromov inequality allows us to bound it from above in terms of the dimension when the Ricci curvature of is nonnegative.
Since the metric measure space has a finite covering constant and a non atomic measure, one can apply Theorem 1.1 of [18] and conclude that there exists a constant depending only on such that for each positive integer , there exists a family of annuli on such that the annuli are mutually disjoint and, ,
(5) 
Since the annuli are mutually disjoint, one has
Thus, at most annuli among satisfy
The corresponding functions are such that (Lemma 2.1)
Using inequality (5), we get
Since the functions are disjointly supported, they form a dimensional subspace on which the Rayleigh quotient is bounded above by the right hand side of the last inequality. We set and conclude using the minmax formula.
As we mentioned above, if is closed and the Ricci curvature of is non negative, then the constants , and, hence, are bounded in terms of the dimension .
Proof of part (II) of Theorem 2.1. Assume now that is a compact orientable surface of genus , possibly with boundary, and let be a positive function on . If has nonempty boundary, then we glue a disk on each boundary component of and extend by . This closed surface admits a conformal branched cover over with degree We endow with the usual spherical distance and the pushforward measure , We apply Theorem 1.1 of [18] to the metric measure space and deduce that there exist an absolute constant and annuli such that the annuli are mutually disjoint and, ,
(6) 
We set for each , . From the conformal invariance of the energy and Lemma 2.1, one deduces that, for every ,
while, since is equal to 1 on ,
Therefore,
where is an absolute constant. Noting that the functions are disjointly supported in , we deduce the desired inequality for .
We end this section with the following observation showing that the presence of the density in the RHS of the inequalities of Theorem 1.1 is essential. This question will also be discussed in Section 5.
Proposition 2.1.
Let be a compact Riemannian manifold and let be a smooth Morse function on . For every we set . If denotes the number of stable critical points of , then there exists such that, ,
Proof.
First observe that for any density , the operator is unitarily equivalent to the Schrödinger operator
acting on (indeed, , where multiplication by is a unitary transform from to ). Consequently, denoting by the eigenvalues of the semiclassical Schrödinger operator , we get
According to [22] (see also [23, proposition 2.2]), there exists such that, , the number of eigenvalues contained in the interval is exactly , that is, and . Therefore,
∎
3. Sharp estimates for the first positive eigenvalue
Let be a compact connected Riemannian dimensional manifold, possibly with a nonempty boundary. Li and Yau introduced in [27] the notion of conformal volume as follows : Given any immersion from to the standard sphere of dimension , we denote by the volume of with respect to the metric , and by the supremum of as runs over the group of conformal diffeomorphisms of . The conformal volume of is
where is the set of all conformal immersions from to .
With the same notations as in the previous section, we have the following
Theorem 3.1.
Let be a compact Riemannian manifold possibly with nonempty boundary. Let be a nonnegative function and let be a nonatomic Radon measure on with . One has
(7) 
with the convention that if .
Proof.
From the definition of , it is clear that if is any nonzero function such that , then, taking for the 2dimensional vector space generated by constant functions and , we have
Let . Using standard center of mass lemma (see e.g., [15, Proposition 4.1.5]), there exists a conformal diffeomorphism of so that the Euclidean components of the map satisfy
Thus, for every ,
(8) 
From the fact that is conformal one has and
We sum up in (8) and use Hölder’s inequality to get
The proof of the theorem follows immediately. ∎
An immediate consequence of Theorem 3.1 is the following
Corollary 3.1.
Let be a compact Riemannian manifold possibly with nonempty boundary and let be a positive function. One has
(9) 
In [8], Ilias and the second author proved that if the Riemannian manifold admits an isometric immersion into a Euclidean space whose Euclidean components are first nonconstant eigenfunctions of the Laplacian , then the following equality holds
(10) 
In this case, the inequality (9) reads
(11) 
where the equality holds whenever is a constant function. This proves the sharpness of the inequality of Corollary 3.1.
Notice that all compact rank one symmetric spaces satisfy (10) and hence (11). In particular, we have the following result that extends Hersch’s isoperimetric inequality [24] and its generalization to higher dimensions [8].
Corollary 3.2.
Let be a Riemannian metric on which is conformal to the standard metric , and let be a positive function. One has
(12) 
where is the volume of the Euclidean unit sphere. Moreover, the equality holds in (12) if and only if is constant and is homothetically equivalent to .
Proof.
Let be conformal transformation of such that, for every , we have
Using the same arguments as in the proof of Theorem 3.1, we get
with (since is conformal from to )
The inequality (12) follows immediately.
Assume that the equality holds in (12). This implies that

, , and

the function is constant if and, if , the functions and are proportional.
An elementary computation gives
From the previous facts we see that is constant in all cases and, since is homothetically equivalent to . ∎ , the metric
4. Eigenvalues of Laplace type operators
In this section we show that Theorem 1.1 extends to a much more general framework to give upper bounds of the eigenvalues of certain operators acting on sections of vector bundles, precisely the Laplacetype operators defined in the introduction. Throughout this section, denotes a compact Riemannian manifold without boundary. We will use the notations introduced in Section 2 and refer to [3, 14] for details on Laplace type operators.
Theorem 4.1.
Let be an operator of Laplace type acting on sections of a Riemannian vector bundle over , and let be an eigensection associated to . One has for all
with and . Thus,
a) If then
where is a constant depending only on the conformal class of .
b) If is a compact, orientable surface of genus then :
where is an absolute constant.
Proof.
As , the quadratic form associated to is given by:
where denotes a generic smooth section. If is any Lipschitz function on , then an integration by parts gives (see [6, Lemma 8])
Now assume that is a first eigensection: . Then we obtain:
for all Lipschitz functions . Let and . Restricting the testsections to sections of type , where is Lipschitz (hence in ) and is a fixed first eigensection, an obvious application of the minmax principle gives:
The remaining part of the theorem is an immediate consequence of the last inequality and Theorem 1.1. ∎
Corollary 4.1.
Assume that a Laplace type operator acting on sections of a Riemannian vector bundle over , admits a first eigensection of constant length. Then, for all
Indeed, when is constant, is nothing but the th eigenvalue of the Laplacian acting on functions. In the particular case where is the Hodge Laplacian acting on forms, Corollary 4.1 says that the existence of a nonzero harmonic form of constant length on leads to