On the growth of eigenfunction averages: microlocalization and geometry
Abstract.
Let be a smooth, compact Riemannian manifold and an normalized sequence of Laplace eigenfunctions, . Given a smooth submanifold of codimension , we find conditions on the pair for which
One such condition is that the set of conormal directions to that are recurrent has measure . In particular, we show that the upper bound holds for any if is surface with Anosov geodesic flow or a manifold of constant negative curvature. The results are obtained by characterizing the behavior of the defect measures of eigenfunctions with maximal averages.
1. Introduction
On a compact Riemannian manifold of dimension we consider sequences of Laplace eigenfunctions solving
In this article, we study the average oscillatory behavior of when restricted to a submanifold . In particular, we seek to understand conditions on the pair under which
(1) 
as , where is the volume measure on induced by the Riemannian metric, and is the codimension of .
We note that the bound
(2) 
holds for any pair [Zel, Corollary 3.3], and is sharp in general. Therefore, we seek to give conditions under which the average is submaximal. Integrals of the form (1), where is a curve, have a long history of study. Good [Good] and Hejhal [Hej] study the case in which is a periodic geodesic in a compact hyperbolic manifold, and prove the bound (2) in that case. The work of Zelditch [Zel] in fact shows that (1) holds for a density one subsequence of eigenvalues. Moreover, one can give explicit polynomial improvements on the error term in (2) for a density one subsequence of eigenfunctions [JZ].
These estimates, however, are not generally satisfied for the full sequence of eigenfunctions and the question of when all eigenfunctions satisfy (1) has been studied recently for the case of curves in surfaces [CS, SXZ, Wym, Wym2] and for submanifolds [Wym3]. Finally, given a hypersurface, the question of which eigenfunctions satisfy (1) was studied in [CGT]. In this article, we address both of these questions, strengthening the results concerning which eigenfunctions can have maximal averages on a given submanifold , and giving weaker conditions on the submanifold that guarantee that (1) holds for all eigenfunctions.
This article improves and extends nearly all existing results regarding averages of eigenfunctions over submanifolds. We recover all conditions guaranteeing that the improved bound (1) holds found in [CS, SXZ, Wym, Wym2, Wym3, GT17, Gdefect, CGT, Berard77, SZ16I, SZ16II]. As far as the authors are aware, these papers contain all previously known conditions ensuring improved averages. Moreover, we give strictly weaker conditions guaranteeing (1) when ; we replace the condition that the set of loop directions has measure zero from [Wym3] with the condition that the set of recurrent directions has measure zero. This allows us to prove that under conditions on including those studied in [Good, Hej, CS, SXZ], the improved bound (1) holds unconditionally with respect to the submanifold . These improvements are possible because the main estimate, Theorem 6, gives explicit bounds on averages over submanifolds which depend only on the microlocalization of a sequence of eigenfunctions in the conormal directions to . This gives a new proof of (2) from [Zel] with explicit control over the constant for high energies. In fact, we characterize those defect measures which may support maximal averages. The estimate requires no assumptions on the geometry of or and is purely local. It is only with this bound in place that we use dynamical arguments to draw conclusions about the pairs supporting eigenfunctions with maximal averages. We note, however, that this paper does not obtain logarithmically improved averages as in [Berard77, SXZ, Wym2].
Recall that all compact, negatively curved Riemannian surfaces have Anosov geodesic flow [Anosov]. One consequence of the results in this paper is the following.
Theorem 1.
Suppose is a compact, Riemannian surface with Anosov geodesic flow and is a smooth curve segment with . Then
as for every sequence of Laplace eigenfunctions. Here denotes the derivative in the normal direction to the curve.
In order to state our more general results we introduce some geometric notation. Let be a closed smooth submanifold of codimension . We denote by the conormal bundle to and we write for the unit conormal bundle of , where the metric is induced from that in . We write for the measure on induced by the Sasaki metric on (see e.g. [Eberlein73]). In particular, if are Fermi coordinates in a tubular neighborhood of , where is identified with , we have
where , , and is the dimensional sphere.
Let with
be the first return time. Define the loop set
and first return map by Next, consider the infinite loop sets
and the recurrent set
where
In what follows we write for the canonical projection map onto , and for the Minkowski box dimension of a set .
Theorem 2.
Let be a smooth, compact Riemannian manifold of dimension . Let be a closed embedded submanifold of codimension , and be a subset with boundary satisfying Suppose
Then
as for every sequence of Laplace eigenfunctions.
Theorem 2 improves on the work of Wyman [Wym3], replacing the measure of the loop set , by that of the recurrent set . Taking to be a single point (i.e. ) also recovers the results of [SoggeTothZelditch]; see Remark 1.
When is a hypersurface, i.e. , we can also study the oscillatory behavior of the normal derivative along .
Theorem 3.
Suppose satisfy the assumptions of Theorem 2 with . Then for every sequence of Laplace eigenfunctions
as .
Theorem 2 allows us to derive substantial conclusions about the geometry of submanifolds supporting eigenfunctions with maximal averages. Indeed, if there exists and a sequence of eigenfunctions for which
then,
Next, we present different geometric conditions on which imply . We recall that strictly negative sectional curvature implies Anosov geodesic flow. Also, both Anosov geodesic flow and nonnegative sectional curvature imply that has no conjugate points.
Theorem 4.
Let be a smooth, compact Riemannian manifold of dimension . Let be a closed embedded submanifold of codimension . Suppose one of the following assumptions holds:

has no conjugate points and has codimension .

has no conjugate points and is a geodesic sphere.

has constant negative curvature.

is a surface with Anosov geodesic flow.

has Anosov geodesic flow and nonpositive curvature, and is totally geodesic.

has Anosov geodesic flow and is a subset that lifts to a horosphere.
Then
In addition, condition 1 implies that
Combining Theorems 2 and 4 gives the following result on the oscillatory behavior of eigenfunctions when restricted to .
Corollary 5.
We conjecture that the conclusions of Theorem 4, and hence also Corollary 5, hold in the case that is a manifold with Anosov geodesic flow of any dimension.
Conjecture.
Let be a manifold of dimension with Anosov geodesic flow and let be a submanifold of codimension . Then
1.1. Semiclassical operators and a quantitative estimate
This section contains the key analytic theorem for controlling submanifold averages (Theorem 6) which, in particular, has Theorems 2 and 3 as corollaries. We control the oscillatory behavior of quasimodes of semiclassical pseudodifferential operators using a quantitative estimate relating averages of quasimodes to the behavior of the associated defect measure. As a consequence, we characterize defect measures for which the corresponding quasimodes may have maximal averages.
We say that a sequence of functions is compactly microlocalized if there exists so that
Also, we say that is a quasimode for if
In addition, for , we say that a submanifold of codimension is conormally transverse for if given such that
we have
(3) 
where is the Hamiltonian vector field associated to .
Let
and consider the Hamiltonian flow
We fix and define for a Borel measure on , the measure on by setting
Remark 2 in [CGT] shows that if is a defect measure associated to a quasimode and is conormally transverse for , then is independent of the choice of . It is then natural to replace the fixed choice of with . In particular, for a defect measure associated to ,
(4) 
for all Borel.
Next, let be the geodesic distance to . That is, . Then, define by
Finally, we write when and are mutually singular measures and let be the volume measure induced on by the Sasaki metric.
Theorem 6.
Let be a smooth, compact Riemannian manifold of dimension and have real valued principal symbol . Suppose that is a closed embedded submanifold of codimension conormally transverse for , and that is a compactly microlocalized quasimode for with defect measure . Let and be so that
Let and with . Then there exists , depending only on and , so that
(5) 
In addition to relating the microlocalization of quasimodes to averages on submanifolds, Theorem 6 gives a quantitative version of the bound (2) proved in [Zel, Corollary 3.3] and generalizes the work of the second author [Gdefect, Theorem 2] to manifolds of any codimension. Note also that the estimate (5) is saturated for every on the round sphere .
Remark 1.
A direct consequence of Theorem 6 is the following.
Theorem 7.
Let be a smooth, compact Riemannian manifold of dimension . Let be a closed embedded submanifold of codimension , and let be a subset with boundary satisfying If is a sequence of eigenfunctions with defect measure so that , then
Theorem 7 strengthens the results of [CGT]. In particular, in [CGT], the measure is said to be conormally diffuse if , which of course implies
We note that Theorem 7 is an immediate consequence of Theorem 6. To see this, first observe that if we take , set , and let satisfy then
for any with on . Next, note that in this setting we have . Hence, if
then by Theorem 6,
To see that any is conormally transverse, observe that if , then . In particular, given there exists for which
1.2. Relation with bounds
Observe that taking in (2), and for some the estimate reads,
(6) 
By Remark 1 the constant can be chosen independent of (and indeed, for small , depending only on the injectivity radius of and dimension of [Gdefect]). Estimates of this form are well known, first appearing in [Ava, Lev, Ho68] (see also [EZB, Chapter 7]), and situations which produce sharp examples for (6) are extensively studied. Many works [Berard77, Is, TZ02, SoggeZelditch, SoggeTothZelditch, SZ16I, SZ16II] have studied connections between growth of norms of eigenfunctions and the global geometry of the manifold . More recently [GT17, Gdefect] examine the relation between defect measures and norms.
This article continues in the spirit of [GT17, Gdefect] and, in particular, taking in Theorem 6 (together with Remark 1) recovers [Gdefect, Theorem 2]. Hence this article also generalizes many of the results of [SoggeZelditch, SoggeTothZelditch, SZ16I, SZ16II] to manifolds of lower codimension. For example taking in Theorem 2 gives the main results of [SoggeTothZelditch] (see also [Gdefect, Corollary 1.2]).
1.3. Manifolds with no focal points or Anosov geodesic flow
In order to prove parts 3, 4, 5 and 6 of Theorem 4, we need to use that the underlying manifold has no focal points or Anosov geodesic flow. We show that these structures allow us to restrict to working on the set of points in at which the tangent space to splits into a sum of bounded and unbounded directions. To make this sentence precise we introduce some notation.
If has no conjugate points, then for any , there exist stable and unstable subspaces so that
and
Moreover, if has no focal points then vary continuously with . (See for example [Eberlein73, Proposition 2.13].)
In what follows we write
We define the mixed and split subsets of respectively by
Then we write
(7) 
where we will use when considering manifolds with Anosov geodesic flow and when considering those with no focal points.
Next, we recall that any manifold with no focal points in which every geodesic encounters a point of negative curvature has Anosov geodesic flow [Eberlein73, Corollary 3.4]. In particular, the class of manifolds with Anosov geodesic flows includes those with negative curvature. We also recall that a manifold with Anosov geodesic flow does not have conjugate points and for all
where are the stable and unstable directions as before. (For other characterizations of manifolds with Anosov geodesic flow, see [Eberlein73, Theorem 3.2], [Eberlein73b].) Moreover, there exists so that for all ,
and the spaces are Hölder continuous in [Anosov].
Theorem 8.
Let be a closed embedded submanifold.
If has no focal points, then
If has Anosov geodesic flow, then
Corollary 9.
Let be a closed embedded submanifold of codimension , and let satisfy . Then if has no focal points and
we have
(8) 
as for every sequence of Laplace eigenfunctions. If instead has Ansov geodesic flow then (8) holds when
Note that if , then since . Indeed, it is not possible to have both and unless and hence . In [Wym, Wym2] the author works with nonpositively curved (and hence having no focal points), and a curve. He then imposes the condition that for all time the curvature of , , avoids two special values determined by the tangent vector to , . He shows that under this condition
If , then the lift of to the universal cover of is tangent to a stable or unstable horosphere at and is equal to the curvature of that horosphere. Since this implies that is stable or unstable, the condition there is that Thus, the condition is the generalization to higher codimensions of that in [Wym, Wym2]. We note that [Wym2] obtains the improved upper bound .
1.4. Organization of the paper
We divide the paper into two major parts. The first part of the paper contains all of the analysis of solutions to . The sections in this part, Section 2 and Section 3, contain the proofs of Theorem 6 and Theorem 3 respectively. The second part of our paper, consists of an analysis of the geodesic flow and in particular a study of the recurrent set of . Theorem 2 is proved in Section 4, and Theorems 4 and 8 are proved in Section 5.
Note that as already explained, Corollary 5 is an immediate consequence of combining Theorems 2 and 4.
Also, Theorem 7 is a direct consequence of Theorem 6 and Corollary 9 is a consequence of Theorem 2 and Theorem 8.
Finally, Theorem 1 is exactly part 4 of Theorem 4.
Acknowledgements. Thanks to Semyon Dyatlov, Patrick Eberlein, Colin Guillarmou, and Gabriel Paternain for several discussions on hyperbolic dynamics. J.G. is grateful to the National Science Foundation for support under the Mathematical Sciences Postdoctoral Research Fellowship DMS1502661.
2. Quantitative estimate: Proof of Theorem 6
In Section 2.1 we present the ground work needed for the proof of Theorem 6. In particular, we state the main technical result, Proposition 10, on which the proof of Theorem 6 hinges. We then divide the proof of Theorem 6 in two parts. Assuming the main technical proposition, we first prove the theorem for the case and in Section 2.2, and then generalize it to any subset in Section 2.3. Finally, Section 2.4 is dedicated to the proof of Proposition 10.
Throughout this section we assume that has principal symbol and is conormally transverse for as defined in (3). We also assume throughout this section that is a compactly microlocalized quasimode for .
2.1. Preliminaries.
Let be a smooth closed submanifold and let be an open neighborhood of described in local coordinates as where these coordinates are chosen so that . The coordinates induce coordinates on with , and where we continue to write . In these coordinates, is cotangent to while is conormal to . Since is conormally transverse for , we may assume, without loss of generality, that with dual coordinates where
Consider the cutoff function with
(9) 
with for all . For consider the symbol
(10) 
where is the Riemannian metric on induced by . Let , where denotes the interior of . We start splitting the period integral as
The same proof as [CGT, Lemma 8] yields that for all
(see also Lemma 12).
Choosing , and using the restriction bound from [BGT], we obtain that
(11) 
We control the integral of using the following lemma. Recall that we write for the Hamiltonian vector field corresponding to and for the associated Hamiltonian flow. To shorten notation, we write
Proposition 10.
Let so that on for some . Let . There exists depending only on and so that
The proof of Proposition 10 is given in Section 2.4. The purpose of this proposition is to allow us to use to localize quasimodes to the support of and its complement. Since and are mutually singular, it is not difficult to see that Proposition 10 gives a bound for of the form . By further restricting to shrinking balls inside an application of the Lebesgue differentiation theorem allows us to obtain a bound of the form as claimed. This improvement will be needed when passing to subsets . The factor measures the cost of restricting to a hypersurface containing which is microlocally transversal to . In particular, we choose coordinates so that and at a point . This is possible since is conormally transverse for .
To apply Proposition 10 it is key to work with cutoff functions so that on for some . Therefore, the following lemma is dedicated to extending cutoff functions on to cutoff functions on that are invariant under the Hamiltonian flow inside . Let be so that
is a diffeomorphism for all . Such a exists since is compact and conormally transverse for . Moreover, for , is a closed embedded submanifold in .
Lemma 11.
For all and there exists
so that
for all and . In particular, on .
Proof.
Let be a fixed function supported on with on . Then, using that is a diffeomorphism, define the smooth cutoff by the relation
Finally, extend to all of so that . We can make such an extension since is a closed embedded submanifold in . ∎
2.2. Proof of Theorem 6 for
Fix . Since and are two Radon measures on that are mutually singular, there exist compact and with and so that
Indeed, by definition of mutual singularity, there exist so that and . Hence, by outer regularity of , there exists open with Next, by inner regularity, of , there exists compact with Let be a cutoff function with
Let be the cutoff extension of