Microlocal asymptotically hyperbolic and Kerr-de Sitter

Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces

András Vasy

with an appendix by Semyon Dyatlov
Department of Mathematics, Stanford University, CA 94305-2125, USA andras@math.stanford.edu
Revised: November 1, 2011. Original version: December 20, 2010
Abstract.

In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of non-elliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Section 2, resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis.

Many natural applications arise in the setting of non-Riemannian b-metrics in the context of Melrose’s b-structures. These include asymptotically de Sitter-type metrics on a blow-up of the natural compactification, Kerr-de Sitter-type metrics, as well as asymptotically Minkowski metrics.

The simplest application is a new approach to analysis on Riemannian or Lorentzian (or indeed, possibly of other signature) conformally compact spaces (such as asymptotically hyperbolic or de Sitter spaces), including a new construction of the meromorphic extension of the resolvent of the Laplacian in the Riemannian case, as well as high energy estimates for the spectral parameter in strips of the complex plane. These results are also available in a follow-up paper which is more expository in nature, [54].

The appendix written by Dyatlov relates his analysis of resonances on exact Kerr-de Sitter space (which then was used to analyze the wave equation in that setting) to the more general method described here.

2000 Mathematics Subject Classification:
Primary 35L05; Secondary 35P25, 58J47, 83C57
A.V.’s contact information: Department of Mathematics, Stanford University, CA 94305-2125, USA. Fax: 1-650-725-4066. Tel: 1-650-723-2226. E-mail: andras@math.stanford.edu

1. Introduction

In this paper we develop a general microlocal framework which in particular allows us to analyze the asymptotic behavior of solutions of the wave equation on asymptotically Kerr-de Sitter and Minkowski space-times, as well as the behavior of the analytic continuation of the resolvent of the Laplacian on so-called conformally compact spaces. This framework is non-perturbative, and works, in particular, for black holes, for relatively large angular momenta (the restrictions come purely from dynamics, and not from methods of analysis of PDE), and also for perturbations of Kerr-de Sitter space, where ‘perturbation’ is only relevant to the extent that it guarantees that the relevant structures are preserved. In the context of analysis on conformally compact spaces, our framework establishes a Riemannian-Lorentzian duality; in this duality the spaces of different signature are smooth continuations of each other across a boundary at which the differential operator we study has some radial points in the sense of microlocal analysis.

Since it is particularly easy to state, and only involves Riemannian geometry, we start by giving a result on manifolds with even conformally compact metrics. These are Riemannian metrics on the interior of a compact manifold with boundary such that near the boundary , with a product decomposition nearby and a defining function , they are of the form

where is a family of metrics on depending on in an even manner, i.e. only even powers of show up in the Taylor series. (There is a much more natural way to phrase the evenness condition, see [28, Definition 1.2].) We also write for the manifold when the smooth structure has been changed so that is a boundary defining function; thus, a smooth function on is even if and only if it is smooth when regarded as a function on . The analytic continuation of the resolvent in this category (but without the evenness condition) was obtained by Mazzeo and Melrose [37], with possibly some essential singularities at pure imaginary half-integers as noticed by Borthwick and Perry [6]. Using methods of Graham and Zworski [26], Guillarmou [28] showed that for even metrics the latter do not exist, but generically they do exist for non-even metrics. Further, if the manifold is actually asymptotic to hyperbolic space (note that hyperbolic space is of this form in view of the Poincaré model), Melrose, Sá Barreto and Vasy [41] showed high energy resolvent estimates in strips around the real axis via a parametrix construction; these are exactly the estimates that allow expansions for solutions of the wave equation in terms of resonances. Estimates just on the real axis were obtained by Cardoso and Vodev for more general conformal infinities [7, 57]. One implication of our methods is a generalization of these results.

Below denotes ‘Schwartz functions’ on , i.e.  functions vanishing with all derivatives at , and is the dual space of ‘tempered distributions’ (these spaces are naturally identified for and ), while is the standard Sobolev space on (corresponding to extension across the boundary, see e.g. [32, Appendix B], where these are denoted by ) and is the standard semiclassical Sobolev space, so for fixed this is the same as ; see [17, 21].

Theorem.

(See Theorem 4.3 for the full statement.) Suppose that is an -dimensional manifold with boundary with an even Riemannian conformally compact metric . Then the inverse of

written as , has a meromorphic continuation from to ,

with poles with finite rank residues. If in addition is non-trapping, then non-trapping estimates hold in every strip , : for ,

(1.1)

If has compact support in , the norm on can be replaced by the norm.

Further, as stated in Theorem 4.3, the resolvent is semiclassically outgoing with a loss of , in the sense of recent results of Datchev and Vasy [15] and [16]. This means that for mild trapping (where, in a strip near the spectrum, one has polynomially bounded resolvent for a compactly localized version of the trapped model) one obtains resolvent bounds of the same kind as for the above-mentioned trapped models, and lossless estimates microlocally away from the trapping. In particular, one obtains logarithmic losses compared to non-trapping on the spectrum for hyperbolic trapping in the sense of [60, Section 1.2], and polynomial losses in strips, since for the compactly localized model this was recently shown by Wunsch and Zworski [60].

For conformally compact spaces, without using wave propagation as motivation, our method is to change the smooth structure, replacing by , conjugate the operator by an appropriate weight as well as remove a vanishing factor of , and show that the new operator continues smoothly and non-degenerately (in an appropriate sense) across , i.e. , to a (non-elliptic) problem which we can analyze utilizing by now almost standard tools of microlocal analysis. These steps are reflected in the form of the estimate (1.1); shows up in the evenness, conjugation due to the presence of , and the two halves of the vanishing factor of being removed in on the left and right hand sides. This approach is explained in full detail in the more expository and self-contained follow-up article, [54].

However, it is useful to think of a wave equation motivation — then -dimensional hyperbolic space shows up (essentially) as a model at infinity inside a backward light cone from a fixed point at future infinity on -dimensional de Sitter space , see [53, Section 7], where this was used to construct the Poisson operator. More precisely, the light cone is singular at , so to desingularize it, consider . After a Mellin transform in the defining function of the front face; the model continues smoothly across the light cone inside the front face of . The inside of the light cone corresponds to -dimensional hyperbolic space (after conjugation, etc.) while the exterior is (essentially) -dimensional de Sitter space; is the ‘boundary’ separating them. Here should be thought of as the event horizon in black hole terms (there is nothing more to event horizons in terms of local geometry!).

The resulting operator has radial points at the conormal bundle of in the sense of microlocal analysis, i.e. the Hamilton vector field is radial at these points, i.e. is a multiple of the generator of dilations of the fibers of the cotangent bundle there. However, tools exist to deal with these, going back to Melrose’s geometric treatment of scattering theory on asymptotically Euclidean spaces [39]. Note that consists of two components, , resp. , and in the images, , resp. , of these are sinks, resp. sources, for the Hamilton flow. At one has choices regarding the direction one wants to propagate estimates (into or out of the radial points), which directly correspond to working with strong or weak Sobolev spaces. For the present problem, the relevant choice is propagating estimates away from the radial points, thus working with the ‘good’ Sobolev spaces (which can be taken to have as positive order as one wishes; there is a minimum amount of regularity imposed by our choice of propagation direction, cf. the requirement above (1.1)). All other points are either elliptic, or real principal type. It remains to either deal with the non-compactness of the ‘far end’ of the -dimensional de Sitter space — or instead, as is indeed more convenient when one wants to deal with more singular geometries, adding complex absorbing potentials, in the spirit of works of Nonnenmacher and Zworski [44] and Wunsch and Zworski [60]. In fact, the complex absorption could be replaced by adding a space-like boundary, see Remark 2.5, but for many microlocal purposes complex absorption is more desirable, hence we follow the latter method. However, crucially, these complex absorbing techniques (or the addition of a space-like boundary) already enter in the non-semiclassical problem in our case, as we are in a non-elliptic setting.

One can reverse the direction of the argument and analyze the wave equation on an -dimensional even asymptotically de Sitter space by extending it across the boundary, much like the the Riemannian conformally compact space is extended in this approach. Then, performing microlocal propagation in the opposite direction, which amounts to working with the adjoint operators that we already need in order to prove existence of solutions for the Riemannian spaces111This adjoint analysis also shows up for Minkowski space-time as the ‘original’ problem., we obtain existence, uniqueness and structure results for asymptotically de Sitter spaces, recovering a large part222Though not the parametrix construction for the Poisson operator, or for the forward fundamental solution of Baskin [1]; for these we would need a parametrix construction in the present compact boundaryless, but analytically non-trivial (for this purpose), setting. of the results of [53]. Here we only briefly indicate this method of analysis in Remark 4.6.

In other words, we establish a Riemannian-Lorentzian duality, that will have counterparts both in the pseudo-Riemannian setting of higher signature and in higher rank symmetric spaces, though in the latter the analysis might become more complicated. Note that asymptotically hyperbolic and de Sitter spaces are not connected by a ‘complex rotation’ (in the sense of an actual deformation); they are smooth continuations of each other in the sense we just discussed.

To emphasize the simplicity of our method, we list all of the microlocal techniques (which are relevant both in the classical and in the semiclassical setting) that we use on a compact manifold without boundary; in all cases only microlocal Sobolev estimates matter (not parametrices, etc.):

  1. Microlocal elliptic regularity.

  2. Real principal type propagation of singularities.

  3. Rough analysis at a Lagrangian invariant under the Hamilton flow which roughly behaves like a collection of radial points, though the internal structure does not matter, in the spirit of [39, Section 9].

  4. Complex absorbing ‘potentials’ in the spirit of [44] and [60].

These are almost ‘off the shelf’ in terms of modern microlocal analysis, and thus our approach, from a microlocal perspective, is quite simple. We use these to show that on the continuation across the boundary of the conformally compact space we have a Fredholm problem, on a perhaps slightly exotic function space, which however is (perhaps apart from the complex absorption) the simplest possible coisotropic function space based on a Sobolev space, with order dictated by the radial points. Also, we propagate the estimates along bicharacteristics in different directions depending on the component of the characteristic set under consideration; correspondingly the sign of the complex absorbing ‘potential’ will vary with , which is perhaps slightly unusual. However, this is completely parallel to solving the standard Cauchy, or forward, problem for the wave equation, where one propagates estimates in opposite directions relative to the Hamilton vector field in the two components.

The complex absorption we use modifies the operator outside . However, while depends on , its behavior on , and even near , is independent of this choice; see the proof of Proposition 4.2 for a detailed explanation. In particular, although may have resonances other than those of , the resonant states of these additional resonances are supported outside , hence do not affect the singular behavior of the resolvent in . In the setting of Kerr-de Sitter space an analogous role is played by semiclassical versions of the standard energy estimate; this is stated in Subsection 3.3.

While the results are stated for the scalar equation, analogous results hold for operators on natural vector bundles, such as the Laplacian on differential forms. This is so because the results work if the principal symbol of the extended problem is scalar with the demanded properties, and the imaginary part of the subprincipal symbol is either scalar at the ‘radial sets’, or instead satisfies appropriate estimates (as an endomorphism of the pull-back of the vector bundle to the cotangent bundle) at this location; see Remark 2.1. The only change in terms of results on asymptotically hyperbolic spaces is that the threshold is shifted; in terms of the explicit conjugation of Subsection 4.9 this is so because of the change in the first order term in (4.29).

While here we mostly consider conformally compact Riemannian or Lorentzian spaces (such as hyperbolic space and de Sitter space) as appropriate boundary values (Mellin transform) of a blow-up of de Sitter space of one higher dimension, they also show up as a boundary value of Minkowski space. This is related to Wang’s work on b-regularity [59], though Wang worked on a blown up version of Minkowski space-time; she also obtained her results for the (non-linear) Einstein equation there. It is also related to the work of Fefferman and Graham [22] on conformal invariants by extending an asymptotically hyperbolic manifold to Minkowski-type spaces of one higher dimension. We discuss asymptotically Minkowski spaces briefly in Section 5.

Apart from trapping — which is well away from the event horizons for black holes that do not rotate too fast — the microlocal structure on de Sitter space is exactly the same as on Kerr-de Sitter space, or indeed Kerr space near the event horizon. (Kerr space has a Minkowski-type end as well; although Minkowski space also fits into our framework, it does so a different way than Kerr at the event horizon, so the result there is not immediate; see the comments below.) This is to be understood as follows: from the perspective we present here (as opposed to the perspective of [53]), the tools that go into the analysis of de Sitter space-time suffice also for Kerr-de Sitter space, and indeed a much wider class, apart from the need to deal with trapping. The trapping itself was analyzed by Wunsch and Zworski [60]; their work fits immediately with our microlocal methods. Phenomena such as the ergosphere are mere shadows of dynamics in the phase space which is barely changed, but whose projection to the base space (physical space) undergoes serious changes. It is thus of great value to work microlocally, although it is certainly possible that for some non-linear purposes it is convenient to rely on physical space to the maximum possible extent, as was done in the recent (linear) works of Dafermos and Rodnianski [13, 14].

Below we state theorems for Kerr-de Sitter space time. However, it is important to note that all of these theorems have analogues in the general microlocal framework discussed in Section 2. In particular, analogous theorems hold on conjugated, re-weighted, and even versions of Laplacians on conformally compact spaces (of which one example was stated above as a theorem), and similar results apply on ‘asymptotically Minkowski’ spaces, with the slight twist that it is adjoints of operators considered here that play the direct role there.

We now turn to Kerr-de Sitter space-time and give some history. In exact Kerr-de Sitter space and for small angular momentum, Dyatlov [20, 19] has shown exponential decay to constants, even across the event horizon. This followed earlier work of Melrose, Sá Barreto and Vasy [40], where this was shown up to the event horizon in de Sitter-Schwarzschild space-times or spaces strongly asymptotic to these (in particular, no rotation of the black hole is allowed), and of Dafermos and Rodnianski in [11] who had shown polynomial decay in this setting. These in turn followed up pioneering work of Sá Barreto and Zworski [47] and Bony and Häfner [5] who studied resonances and decay away from the event horizon in these settings. (One can solve the wave equation explicitly on de Sitter space using special functions, see [45] and [61]; on asymptotically de Sitter spaces the forward fundamental solution was constructed as an appropriate Lagrangian distribution by Baskin [1].)

Also, polynomial decay on Kerr space was shown recently by Tataru and Tohaneanu [50, 49] and Dafermos and Rodnianski [13, 14], after pioneering work of Kay and Wald in [33] and [58] in the Schwarzschild setting. (There was also recent work by Marzuola, Metcalf, Tataru and Tohaneanu [36] on Strichartz estimates, and by Donninger, Schlag and Soffer [18] on estimates on Schwarzschild black holes, following estimates of Dafermos and Rodnianski [12, 10], of Blue and Soffer [4] on non-rotating charged black holes giving estimates, and Finster, Kamran, Smoller and Yau [23, 24] on Dirac waves on Kerr.) While some of these papers employ microlocal methods at the trapped set, they are mostly based on physical space where the phenomena are less clear than in phase space (unstable tools, such as separation of variables, are often used in phase space though). We remark that Kerr space is less amenable to immediate microlocal analysis to attack the decay of solutions of the wave equation due to the singular/degenerate behavior at zero frequency, which will be explained below briefly. This is closely related to the behavior of solutions of the wave equation on Minkowski space-times. Although our methods also deal with Minkowski space-times, this holds in a slightly different way than for de Sitter (or Kerr-de Sitter) type spaces at infinity, and combining the two ingredients requires some additional work. On perturbations of Minkowski space itself, the full non-linear analysis was done in the path-breaking work of Christodoulou and Klainerman [9], and Lindblad and Rodnianski simplified the analysis [34, 35], Bieri [2, 3] succeeded in relaxing the decay conditions, while Wang [59] obtained additional, b-type, regularity as already mentioned. Here we only give a linear result, but hopefully its simplicity will also shed new light on the non-linear problem.

As already mentioned, a microlocal study of the trapping in Kerr or Kerr-de Sitter was performed by Wunsch and Zworski in [60]. This is particularly important to us, as this is the only part of the phase space which does not fit directly into a relatively simple microlocal framework. Our general method is to use microlocal analysis to understand the rest of the phase space (with localization away from trapping realized via a complex absorbing potential), then use the gluing result of Datchev and Vasy [15] to obtain the full result.

Slightly more concretely, in the appropriate (partial) compactification of space-time, near the boundary of which space-time has the form , where denotes an extension of the space-time across the event horizon. Thus, there is a manifold with boundary , whose boundary is the event horizon, such that is embedded into , a (non-compact) manifold without boundary. We write for ‘our side’ of the event horizon and for the ‘far side’. Then the Kerr or Kerr-de Sitter d’Alembertians are b-operators in the sense of Melrose [43] that extend smoothly across the event horizon . Recall that in the Riemannian setting, b-operators are usually called ‘cylindrical ends’, see [43] for a general description; here the form at the boundary (i.e. ‘infinity’) is similar, modulo ellipticity (which is lost). Our results hold for small smooth perturbations of Kerr-de Sitter space in this b-sense. Here the role of ‘perturbations’ is simply to ensure that the microlocal picture, in particular the dynamics, has not changed drastically. Although b-analysis is the right conceptual framework, we mostly work with the Mellin transform, hence on manifolds without boundary, so the reader need not be concerned about the lack of familiarity with b-methods. However, we briefly discuss the basics in Section 3.

We immediately Mellin transform in the defining function of the boundary (which is temporal infinity, though is not space-like everywhere) — in Kerr and Kerr-de Sitter spaces this is operation is ‘exact’, corresponding to being a Killing vector field, i.e. is not merely at the level of normal operators, but this makes little difference (i.e. the general case is similarly treatable). After this transform we get a family of operators that e.g. in de Sitter space is elliptic on , but in Kerr space ellipticity is lost there. We consider the event horizon as a completely artificial boundary even in the de Sitter setting, i.e. work on a manifold that includes a neighborhood of , hence a neighborhood of the event horizon .

As already mentioned, one feature of these space-times is some relatively mild trapping in ; this only plays a role in high energy (in the Mellin parameter, ), or equivalently semiclassical (in ) estimates. We ignore a (semiclassical) microlocal neighborhood of the trapping for a moment; we place an absorbing ‘potential’ there. Another important feature of the space-times is that they are not naturally compact on the ‘far side’ of the event horizon (inside the black hole), i.e. , and bicharacteristics from the event horizon (classical or semiclassical) propagate into this region. However, we place an absorbing ‘potential’ (a second order operator) there to annihilate such phenomena which do not affect what happens on ‘our side’ of the event horizon, , in view of the characteristic nature of the latter. This absorbing ‘potential’ could easily be replaced by a space-like boundary, in the spirit of introducing a boundary , where , when one solves the Cauchy problem from for the standard wave equation; note that such a boundary does not affect the solution of the equation in . Alternatively, if has a well-behaved infinity, such as in de Sitter space, the analysis could be carried out more globally. However, as we wish to emphasize the microlocal simplicity of the problem, we do not touch on these issues.

All of our results are in a general setting of microlocal analysis explained in Section 2, with the Mellin transform and Lorentzian connection explained in Section 3. However, for the convenience of the reader here we state the results for perturbations of Kerr-de Sitter spaces. We refer to Section 6 for details. First, the general assumption is that

, , is either the Mellin transform of the d’Alembertian for a Kerr-de Sitter spacetime, or more generally the Mellin transform of the normal operator of the d’Alembertian for a small perturbation, in the sense of b-metrics, of such a Kerr-de Sitter space-time;

see Section 3 for an explanation of these concepts. Note that for such perturbations the usual ‘time’ Killing vector field (denoted by in Section 6; this is indeed time-like in sufficiently far from ) is no longer Killing. Our results on these space-times are proved by showing that the hypotheses of Section 2 are satisfied. We show this in general (under the conditions (6.2), which corresponds to in de Sitter-Schwarzschild spaces, and (6.12), which corresponds to the lack of classical trapping in ; see Section 6), except where semiclassical dynamics matters. As in the analysis of Riemannian conformally compact spaces, we use a complex absorbing operator ; this means that its principal symbol in the relevant (classical, or semiclassical) sense has the correct sign on the characteristic set; see Section 2.

When semiclassical dynamics does matter, the non-trapping assumption with an absorbing operator , , is

in both the forward and backward directions, the bicharacteristics from any point in the semiclassical characteristic set of either enter the semiclassical elliptic set of at some finite time, or tend to ;

see Definition 2.13. Here, as in the discussion above, are two components of the image of in . (As is a sink while is a source, even semiclassically, outside the ‘tending’ can only happen in the forward, resp. backward, directions.) Note that the semiclassical non-trapping assumption (in the precise sense used below) implies a classical non-trapping assumption, i.e. the analogous statement for classical bicharacteristics, i.e. those in . It is important to keep in mind that the classical non-trapping assumption can always be satisfied with supported in , far from .

In our first result in the Kerr-de Sitter type setting, to keep things simple, we ignore semiclassical trapping via the use of ; this means that will have support in . However, in , only matters in the semiclassical, or high energy, regime, and only for (almost) real . If the black hole is rotating relatively slowly, e.g.  satisfies the bound (6.22), the (semiclassical) trapping is always far from the event horizon, and one can make supported away from there. Also, the Klein-Gordon parameter below is ‘free’ in the sense that it does not affect any of the relevant information in the analysis333It does affect the location of the poles and corresponding resonant states of , hence the constant in Theorem 1.4 has to be replaced by the appropriate resonant state and exponential growth/decay, as in the second part of that theorem. (principal and subprincipal symbol; see below). Thus, we drop it in the following theorems for simplicity.

Theorem 1.1.

Let be an absorbing formally self-adjoint operator such that the semiclassical non-trapping assumption holds. Let , and

Let be given by the geometry at conormal bundle of the black hole (), resp. de Sitter () event horizons, see Subsection 6.1, and in particular (6.9). For , let444This means that we require the stronger of to hold in (1.2). If we perturb Kerr-de Sitter space time, we need to increase the requirement on slightly, i.e. the size of the half space has to be slightly reduced. if , if . Then, for ,

is an analytic family of Fredholm operators on

(1.2)

and has a meromorphic inverse,

which is holomorphic in an upper half plane, . Moreover, given any , there are only finitely many poles in , and the resolvent satisfies non-trapping estimates there, which e.g. with (which might need a reduction in ) take the form

The analogous result also holds on Kerr space-time if we suppress the Euclidean end by a complex absorption.

Dropping the semiclassical absorption in , i.e. if we make supported only in , we have555Since we are not making a statement for almost real , semiclassical trapping, discussed in the previous paragraph, does not matter.

Theorem 1.2.

Let , , be as in Theorem 1.1, and let be an absorbing formally self-adjoint operator supported in which is classically non-trapping. Let , and

with

Then,

is an analytic family of Fredholm operators on , and has a meromorphic inverse,

which for any is holomorphic in a translated sector in the upper half plane, . The poles of the resolvent are called resonances. In addition, taking for instance, satisfies non-trapping estimates, e.g. with ,

in such a translated sector.

It is in this setting that could be replaced by working on a manifold with boundary, with the boundary being space-like, essentially as a time level set mentioned above, since it is supported in .

Now we make the assumption that the only semiclassical trapping is due to hyperbolic trapping with trapped set , , with hyperbolicity understood as in the ‘Dynamical Hypotheses’ part of [60, Section 1.2], i.e.

in both the forward and backward directions, the bicharacteristics from any point in the semiclassical characteristic set of either enter the semiclassical elliptic set of at some finite time, or tend to .

We remark that just hyperbolicity of the trapped set suffices for the results of [60], see Section 1.2 of that paper; however, if one wants stability of the results under perturbations, one needs to assume that is normally hyperbolic. We refer to [60, Section 1.2] for a discussion of these concepts. We show in Section 6 that for black holes satisfying (6.22) (so the angular momentum can be comparable to the mass) the operators can be chosen so that they are supported in (even quite far from ) and the hyperbolicity requirement is satisfied. Further, we also show that for slowly rotating black holes the trapping is normally hyperbolic. Moreover, the (normally) hyperbolic trapping statement is purely in Hamiltonian dynamics, not regarding PDEs. It might be known for an even larger range of rotation speeds, but the author is not aware of this.

Under this assumption, one can combine Theorem 1.1 with the results of Wunsch and Zworski [60] about hyperbolic trapping and the gluing results of Datchev and Vasy [15] to obtain a better result for the merely spatially localized problem, Theorem 1.2:

Theorem 1.3.

Let , , , , and be as in Theorem 1.2, and assume that the only semiclassical trapping is due to hyperbolic trapping. Then,

is an analytic family of Fredholm operators on , and has a meromorphic inverse,

which is holomorphic in an upper half plane, . Moreover, there exists such that there are only finitely many poles in , and the resolvent satisfies polynomial estimates there as , , for some , compared to the non-trapping case, with merely a logarithmic loss compared to non-trapping for real , e.g. with :

Farther, there are approximate lattices of poles generated by the trapping, as studied by Sá Barreto and Zworski in [47], and further by Bony and Häfner in [5], in the exact De Sitter-Schwarzschild and Schwarzschild settings, and in ongoing work by Dyatlov in the exact Kerr-de Sitter setting.

Theorem 1.3 immediately and directly gives the asymptotic behavior of solutions of the wave equation across the event horizon. Namely, the asymptotics of the wave equation depends on the finite number of resonances; their precise behavior depends on specifics of the space-time, i.e. on these resonances. This is true even in arbitrarily regular b-Sobolev spaces – in fact, the more decay we want to show, the higher Sobolev spaces we need to work in. Thus, a forteriori, this gives estimates. We state this formally as a theorem in the simplest case of slow rotation; in the general case one needs to analyze the (finite!) set of resonances along the reals to obtain such a conclusion, and for the perturbation part also to show normal hyperbolicity (which we only show for slow rotation):

Theorem 1.4.

Let be the partial compactification of Kerr-de Sitter space as in Section 6, with the boundary defining function. Suppose that is either a slowly rotating Kerr-de Sitter metric, or a small perturbation as a symmetric bilinear form on . Then there exist , such that for and solutions of with vanishing in , and with vanishing in , satisfy that for some constant ,

Here the norms of both in and in are bounded by that of in for .

More generally, if is a Kerr-de Sitter metric with hyperbolic trapping666This is shown in Section 6 when (6.22) is satisfied., then there exist , such that for and solutions of with vanishing in , and with vanishing in , satisfy that for some (which are resonant states) and (which are the resonances),

Here the (semi)norms of both in and in are bounded by that of in for . The same conclusion holds for sufficiently small perturbations of the metric as a symmetric bilinear form on provided the trapping is normally hyperbolic.

In special geometries (without the ability to add perturbations) such decay has been described by delicate separation of variables techniques, again see Bony-Häfner [5] in the De Sitter-Schwarzschild and Schwarzschild settings, but only away from the event horizons, and by Dyatlov [20, 19] in the Kerr-de Sitter setting. Thus, in these settings, we recover in a direct manner Dyatlov’s result across the event horizon [19], modulo a knowledge of resonances near the origin contained in [20]. In fact, for small angular momenta one can use the results from de Sitter-Schwarzschild space directly to describe these finitely many resonances, as exposed in the works of Sá Barreto and Zworski [47], Bony and Häfner [5] and Melrose, Sá Barreto and Vasy [40], since is an isolated resonance with multiplicity and eigenfunction ; this persists under small deformations, i.e. for small angular momenta. Thus, exponential decay to constants, Theorem 1.4, follows immediately.

One can also work with Kerr space-time, apart from issues of analytic continuation. By using weighted spaces and Melrose’s results from [39] as well as those of Vasy and Zworski in the semiclassical setting [56], one easily gets an analogue of Theorem 1.2 in , with smoothness and the almost non-trapping estimates corresponding to those of Wunsch and Zworski [60] down to for large. Since a proper treatment of this would exceed the bounds of this paper, we refrain from this here. Unfortunately, even if this analysis were carried out, low energy problems would still remain, so the result is not strong enough to deduce the wave expansion. As already alluded to, Kerr space-time has features of both Minkowski and de Sitter space-times; though both of these fit into our framework, they do so in different ways, so a better way of dealing with the Kerr space-time, namely adapting our methods to it, requires additional work.

While de Sitter-Schwarzschild space (the special case of Kerr-de Sitter space with vanishing rotation), via the same methods as those on de Sitter space which give rise to the hyperbolic Laplacian and its continuation across infinity, gives rise essentially to the Laplacian of a conformally compact metric, with similar structure but different curvature at the two ends (this was used by Melrose, Sá Barreto and Vasy [40] to do analysis up to the event horizon there), the analogous problem for Kerr-de Sitter is of edge-type in the sense of Mazzeo’s edge calculus [38] apart from a degeneracy at the poles corresponding to the axis of rotation, though it is not Riemannian. Note that edge operators have global properties in the fibers; in this case these fibers are the orbits of rotation. A reasonable interpretation of the appearance of this class of operators is that the global properties in the fibers capture non-constant (or non-radial) bicharacteristics (in the classical sense) in the conormal bundle of the event horizon, and also possibly the (classical) bicharacteristics entering . This suggests that the methods of Melrose, Sá Barreto and Vasy [40] would be much harder to apply in the presence of rotation.

It is important to point out that the results of this paper are stable under small perturbations777Certain kinds of perturbations conormal to the boundary, in particular polyhomogeneous ones, would only change the analysis and the conclusions slightly. of the Lorentzian metric on the b-cotangent bundle at the cost of changing the function spaces slightly; this follows from the estimates being stable in these circumstances. Note that the function spaces depend on the principal symbol of the operator under consideration, and the range of depends on the subprincipal symbol at the conormal bundle of the event horizon; under general small smooth perturbations, defining the spaces exactly as before, the results remain valid if the range of is slightly restricted.

In addition, the method is stable under gluing: already Kerr-de Sitter space behaves as two separate black holes (the Kerr and the de Sitter end), connected by semiclassical dynamics; since only one component (say ) of the semiclassical characteristic set moves far into , one can easily add as many Kerr black holes as one wishes by gluing beyond the reach of the other component, . Theorems 1.1 and 1.2 automatically remain valid (for the semiclassical characteristic set is then irrelevant), while Theorem 1.3 remains valid provided that the resulting dynamics only exhibits mild trapping (so that compactly localized models have at most polynomial resolvent growth), such as normal hyperbolicity, found in Kerr-de Sitter space.

Since the specifics of Kerr-de Sitter space-time are, as already mentioned, irrelevant in the microlocal approach we take, we start with the abstract microlocal discussion in Section 2, which is translated into the setting of the wave equation on manifolds with a Lorentzian b-metric in Section 3, followed by the description of de Sitter, Minkowski and Kerr-de Sitter space-times in Sections 4, 5 and 6. Theorems 1.1-1.4 are proved in Section 6 by showing that they fit into the abstract framework of Section 2; the approach is completely analogous to de Sitter and Minkowski spaces, where the fact that they fit into the abstract framework is shown in Sections 4 and 5. As another option, we encourage the reader to read the discussion of de Sitter space first, which also includes the discussion of conformally compact spaces, presented in Section 4, as well as Minkowski space-time presented in the section afterwards, to gain some geometric insight, then the general microlocal machinery, and finally the Kerr discussion to see how that space-time fits into our setting. Finally, if the reader is interested how conformally compact metrics fit into the framework and wants to jump to the relevant calculation, a reasonable place to start is Subsection 4.9. We emphasize that for the conformally compact results, only Section 2 and Section 4.4-4.9, starting with the paragraph of (4.8), are strictly needed.

2. Microlocal framework

We now develop a setting which includes the geometry of the ‘spatial’ model of de Sitter space near its ‘event horizon’, as well as the model of Kerr and Kerr-de Sitter settings near the event horizon, and the model at infinity for Minkowski space-time near the light cone (corresponding to the adjoint of the problem described below in the last case). As a general reference for microlocal analysis, we refer to [32], while for semiclassical analysis, we refer to [17, 21]; see also [48] for the high-energy (or large parameter) point of view.

2.1. Notation

We recall the basic conversion between these frameworks. First, is the set of functions on satisfying uniform bounds

If and are open, we define by requiring888Another possibility would be to require uniform estimates on compact subsets; this makes no difference here. these estimates to hold only for and . The class of classical (or one-step polyhomogeneous) symbols is the subset of consisting of symbols possessing an asymptotic expansion

where . Then on , pseudodifferential operators are of the form

understood as an oscillatory integral. Classical pseudodifferential operators, , form the subset where is a classical symbol. The principal symbol of is the equivalence class of in . For classical , one can instead consider as the principal symbol; it is a function on , which is homogeneous of degree with respect to the -action given by dilations in the second factor, .

Differential operators on form the subset of in which is polynomial in the second factor, , so locally

If is a manifold, one can transfer these definitions to by localization and requiring that the Schwartz kernels are densities away from the diagonal in ; then is in , resp.  when , resp. ; here is the zero section, and stands for symbols homogeneous with respect to the action. If is a differential operator, then the classical (i.e. homogeneous) version of the principal symbol is a homogeneous polynomial in the fibers of the cotangent bundle of degree . We can also work with operators depending on a parameter by replacing by , with smoothly dependent on . In the case of differential operators, would simply depend smoothly on the parameter .

The large parameter, or high energy, version of this, with the large parameter denoted by , is that

where , with identified with ; thus there are joint symbol estimates in and . The high energy principal symbol now should be thought of as an equivalence class of functions on , or invariantly on . Differential operators with polynomial dependence on now take the form

(2.1)

Note that the principal symbol includes terms that would be subprincipal with considered as a differential operator for a fixed value of .

The semiclassical operator algebra999We adopt the convention that denotes semiclassical objects, while is the actual semiclassical parameter., , is given by

its classical subalgebra, corresponds to . The semiclassical principal symbol is now . We can again add an extra parameter , so ; then in the invariant setting the principal symbol is . Note that if is a classical operator with a large parameter, then for , compact, ,

and . The converse is not quite true: roughly speaking, the semiclassical algebra is a blow-up of the large parameter algebra; to obtain an equivalence, we would need to demand in the definition of the large parameter algebra merely that , so in particular for bounded , is merely a family of symbols depending smoothly on (not jointly symbolic); we do not discuss this here further. Note, however, that it is the (smaller, i.e. stronger) large parameter algebra that arises naturally when one Mellin transforms in the b-setting, see Subsection 3.1.

Differential operators now take the form

(2.2)

Such a family has two principal symbols, the standard one (but taking into account the semiclassical degeneration, i.e. based on rather than ), which depends on and is homogeneous, and the semiclassical one, which is at , and is not homogeneous:

However, the restriction of to is the principal part of . In the special case in which is independent of (which is true in the setting considered below), one can simply regard the usual principal symbol as the principal part of the semiclassical symbol. Note that for as in (2.1),

which is indeed of the form (2.2), with polynomial dependence on both and . Note that in this case the standard principal symbol is independent of and .

2.2. General assumptions

Let be a compact manifold and a smooth non-vanishing density on it; thus is well-defined as a Hilbert space (and not only up to equivalence). We consider operators on depending on a complex parameter , with the dependence being analytic (thus, if is a differential operator, the coefficients depend analytically on ). We also consider a complex absorbing ‘potential’, , defined for , is open. It can be convenient to take formally self-adjoint, which is possible when is independent of , but this is inconvenient when one wants to study the large (i.e. semiclassical) behavior. The operators we study are and ; depends on the choice of the density .

Typically we shall be interested in on an open subset of , and have supported in the complement of , such that over some subset of , is elliptic on the characteristic set of . In the Kerr-de Sitter setting, we would have . However, this is not part of the general set-up.

It is often convenient to work with the fiber-radial compactification of , in particular when discussing semiclassical analysis; see for instance [39, Sections 1 and 5]. Thus, should be considered as the boundary of . When one is working with homogeneous objects, as is the case in classical microlocal analysis, one can think of as , but this is not a useful point of view in semiclassical analysis101010In fact, even in classical microlocal analysis it is better to keep at least a ‘shadow’ of the interior of by working with considered as a half-line bundle over with homogeneous objects on it; this keeps the action of the Hamilton vector field on the fiber-radial variable, i.e. the defining function of in , non-trivial, which is important at radial points.. Thus, if is a non-vanishing homogeneous degree function on , it is a defining function of in ; if the homogeneity requirement is dropped it can be modified near the zero section to make it a defining function of in . The principal symbols of are homogeneous degree functions on , so are homogeneous degree there, thus are functions111111This depends on choices unless ; they are naturally sections of a line bundle that encodes the differential of the boundary defining function at . However, the only relevant notion here is ellipticity, and later the Hamilton vector field up to multiplication by a positive function, which is independent of choices. In fact, we emphasize that all the requirements listed for , and later and , except possibly (2.5)-(2.6), are also fulfilled if is replaced by any smooth positive multiple, so one may factor out positive factors at will. This is useful in the Kerr-de Sitter space discussion. For (2.5)-(2.6), see Footnote 14. on near its boundary, , and in particular on . Moreover, is homogeneous degree on , thus a smooth vector field tangent to the boundary on (defined near the boundary), and in particular induces a smooth vector field on .

We assume that the principal symbol , resp. , of , resp. , are real, are independent of , implies . We assume that the characteristic set of is of the form

are relatively open121212Thus, they are connected components in the extended sense that they may be empty. in , and

We assume that there are conic submanifolds of , outside which the Hamilton vector field is not radial, and to which the Hamilton vector field is tangent. Here are typically Lagrangian, but this is not needed131313An extreme example would be . Another extreme is if one or both are empty.. The properties we want at are (probably) not stable under general smooth perturbations; the perturbations need to have certain properties at . However, the estimates we then derive are stable under such perturbations. First, we want that for a homogeneous degree defining function of near , the image of in ,

(2.3)

Next, we require the existence of a non-negative homogeneous degree zero quadratic defining function , of (i.e. it vanishes quadratically at , and is non-degenerate) and such that

(2.4)

is modulo cubic vanishing terms at . (The precise behavior of , or of linear defining functions, is irrelevant, because we only need a relatively weak estimate. It would be relevant if one wanted to prove Lagrangian regularity.) Under these assumptions, is a source and is a sink for the -dynamics in the sense that nearby bicharacteristics tend to as the parameter along the bicharacteristic goes to . Finally, we assume that the imaginary part of the subprincipal symbol at , which is the symbol of as is real, is141414If is radial at , this is independent of the choice of the density . Indeed, with respect to , the adjoint of is , with denoting the adjoint with respect to . This is , and the principal symbol of vanishes at as . In general, we can only change the density by factors with , which in Kerr-de Sitter space-times would mean factors independent of at the event horizon. A similar argument shows the independence of the condition from the choice of when one replaces by , under the same conditions: either radiality, or just .

(2.5)

is positive along , and write

(2.6)

If is a constant, we may write

(2.7)

The results take a little nicer form in this case since depending on various signs, sometimes and sometimes is the relevant quantity.

Figure 1. The components of the characteristic set in the cosphere bundle . The submanifolds are points here, with a source, a sink. The thin lined parabolic regions near the edges show the absorbing region, i.e. the support of . For ,the estimates are always propagated away from towards the support of , so in the direction of the Hamilton flow in , and in the direction opposite of the Hamilton flow in ; for , the directions are reversed.

We make the following non-trapping assumption. For , let , resp.  denote the image of the forward, resp. backward, half-bicharacteristic from . We write (and say tends to ) if given any neighborhood of , ; by the source/sink property this implies that the points on the curve are in for sufficiently large (in absolute value) parameter values. We assume that, with denoting the elliptic set of ,

(2.8)

That is, all forward and backward half-(null)bicharacteristics of either enter the elliptic set of , or go to , i.e.  in . The point of the assumptions regarding and the flow is that we are able to propagate estimates forward near where , backward near where , so by our hypotheses we can always propagate estimates for from towards the elliptic set of , and also if both ends of a bicharacteristic go to the elliptic set of then we can propagate the estimates from one of the directions. On the other hand, for , we can propagate estimates from the elliptic set of towards , and again if both ends of a bicharacteristic go to the elliptic set of then we can propagate the estimates from one of the directions. This behavior of vs.  is important for duality reasons.

Remark 2.1.

For simplicity of notation we have not considered vector bundles on . However, if is a vector bundle on with a positive definite inner product on the fibers and with scalar principal symbol , and in case of the imaginary part of the subprincipal symbol is of the form (2.5) with a bundle-endomorphism satisfying an inequality in (2.6) as a bundle endomorphism, the arguments we present go through.

2.3. Elliptic and real principal type points

We now turn to analysis. First, by the usual elliptic theory, on the elliptic set of , so both on the elliptic set of and on the elliptic set of , one has elliptic estimates151515Our convention in estimates such as (2.9) and (2.10) is that if one assumes that all the quantities on the right hand side are in the function spaces indicated by the norms then so is the quantity on the left hand side, and the estimate holds. As we see below, at not all relevant function space statements appear in the estimate, so we need to be more explicit there.: for all and , and for all with elliptic on ,

(2.9)

with the estimate also holding for