The Wave Kernel on Asymptotically Complex Hyperbolic Manifolds

There is a long-standing research program investigating the spectral and scattering theory of real asymptotically hyperbolic manifolds, see e.g. [An10, AlMa10, AlBaNa20, ChDeLeSk05, FeGr85, GrWi99, GrZw01, JoSá00, Va17] and references contained therein, for a small sample of the surrounding work. However there is comparatively much less work concerning the analogous setting of asymptotically complex hyperbolic manifolds. These spaces were first introduced by Epstein, Mendoza, and Melrose [EpMeMe91], and more recently have been investigated extensively by [GuSa06, FeHi03, GuSa06, HMM17, PeHiTa08, Ma16, Ma18]. This class of manifolds includes certain quotients of complex hyperbolic space by discrete groups, as well as strictly pseudoconvex domains in Stein manifolds equipped with Kähler metrics of Bergman type.

In this work we extend major results which study the wave kernel of asymptotically real hyperbolic manifolds to this complex setting. Joshi-Sá Barreto [JoSá01] study the wave kernel by exhibiting this operator as an element of a certain algebra of Fourier integral operators which have been adapted to the geometry at infinity of this class of real asymptotically hyperbolic manifolds. In the case of both works, moving from the real to the complex case presents new difficulties to the analysis. On the other hand, the original methods of both Vasy and Joshi-Sá Barreto are robust enough to permit an analysis of this class of manifolds of hyperbolic-type.

Before introducing the structure of complex hyperbolic manifolds we briefly recall the geometry of real asymptotically hyperbolic manifolds. A non-compact Riemannian manifold $(M,g)$ of real dimension $(n+1)$ is called asymptotically hyperbolic if it compactifies to a ${\mathcal{C}}^{\infty}$ manifold $\overline{M}$ with compact boundary $\partial\overline{M}$, equipped with a boundary defining function $\rho$, and such that $\rho^{2}g$ is a ${\mathcal{C}}^{\infty}$ metric which is non-degenerate up to the boundary, and moreover that $|d\rho|_{\rho^{2}g}^{2}\equiv 1$ at $\partial\overline{M}$. This name is due to the fact that the final hypothesis ensures that along any smooth curve in $\overline{M}\setminus\partial\overline{M}$ approaching a point in $\partial\overline{M}$, all sectional curvatures of $g$ approach $-1$, see e.g. [MaMe87].

As proven in [JoSá00], these geometric hypotheses are equivalent to the existence of a product-type decomposition at infinity $M\sim[0,\varepsilon)_{\rho}\times\partial M$, such that

where $g_{0}(\rho)$ is a ${\mathcal{C}}^{\infty}$ 1-parameter family of ${\mathcal{C}}^{\infty}$ metrics on $\partial\overline{M}$. In this model, the boundary $\partial\overline{M}$ represents the geometric infinity of $\overline{M}$, analogous to the role played by the $\mathbb{S}^{n}$ at infinity in $\mathbb{H}_{\mathbb{R}}^{n+1}$. In particular the metric $\rho^{2}g|_{\partial M}$ fixes a conformal representative of a metric on $\partial\overline{M}$.

The spectrum of the Laplacian of such manifolds was first studied by [MaMe87]; they determined that it is comprised of finitely many $L^{2}$-eigenvalues $\sigma_{\text{pp}}(\Delta_{g})\subset(0,\tfrac{(n+1)^{2}}{4})$ and the absolutely continuous spectrum $\sigma_{\text{ac}}(\Delta_{g})=[\tfrac{(n+1)^{2}}{4},\infty)$. In particular, they prove that the resolvent

is well-defined as a bounded operator on $L_{g}^{2}(X)$ whenever $\operatorname{Re}(\zeta)>\tfrac{n+1}{2}$. Further they prove that $R(\zeta)$ has a meromorphic extension to $\mathbb{C}\setminus\tfrac{1}{2}((n+1)-\mathbb{N}_{0})$, as an operator $R(\zeta):{\mathcal{C}}^{\infty}_{0}(X)\to{\mathcal{C}}^{\infty}(X)$, and with only finite order poles (this extension is meromorphic on the whole complex plane assuming the metric is even in the sense of [Gu05]).

We now move to introducing the complex analogue of these spaces, and introduce our results. We say a non-compact Riemannian manifold $(X,g)$, of complex dimension $(n+1)$, is an asymptotically complex hyperbolic manifold (hereafter ACH manifold) if the following holds. We assume $X$ compactifies to a ${\mathcal{C}}^{\infty}$ manifold $\overline{X}$, compact with boundary, equipped with a choice of boundary defining function $r$ (hereafter, a bdf). This is a smooth nonnegative function on $\overline{X}$ which such that

We further assume the boundary admits: (1) a contact form $\theta\in\Omega^{1}(\partial\overline{X})$ defined as satisfying $\theta\wedge(d\theta)^{n}\neq 0$; (2) an almost complex structure $J:\operatorname{Ker}\theta\to\operatorname{Ker}\theta$; such that $d\theta(\cdot,J\cdot)$ is a symmetric, positive-definite bilinear form on $\operatorname{Ker}\theta$. Then we say $(X,g)$ is an ACH manifold if there is a tubular neighborhood $\Phi:U\to\partial\overline{X}\times[0,\varepsilon)_{r}$ of the boundary $\partial\overline{X}$ such that

In particular, for another choice of boundary defining function, $\widetilde{r}$, we observe that $r^{4}g|_{\partial\overline{X}}=e^{4f}\theta$, for some $f\in{\mathcal{C}}^{\infty}(\overline{X})$. Denoting the conformal class of our contact structure by $[\theta]$ we can consider the boundary as being endowed with the structure of a conformal pseudohermitian manifold $(\partial\overline{X},[\theta],J)$. This is analogous to the natural conformal structure on $(\partial\overline{M},[\rho^{2}g])$ in the real hyperbolic case.

Before continuing, we require an additional hypothesis, which is that $g$ is an even metric; i.e., the dual metric $g^{-1}$ defined on $T^{*}\overline{X}$ has only even powers of $r$ in a Taylor expansion at $r=0$. This is automatic in the case of $\mathbb{H}_{\mathbb{C}}^{n+1}$, and necessary for the existence of a meromorphic continuation of the resolvent of $\Delta_{g}$ to all of $\mathbb{C}$, (in fact, the failure of this hypothesis implies the existence of at least one essential singularity in the continuation of the resolvent, see [Gu05], [GuSa06]).

In the case that the metric of $(X,g)$ is even in the above sense, we can replace the smooth structure on this manifold with its even smooth structure, denoted $X_{\operatorname{even}}$. In this case the smooth structure on $X$ has been modified by declaring that only functions which are even in $r$ are smooth with respect to $X_{\operatorname{even}}$. This change of the smooth structure permits us to define a square root of our original defining function, and guarantee that it is an element of ${\mathcal{C}}^{\infty}(X_{\operatorname{even}})$. Equivalently, the even smooth structure can be defined by declaring $X_{\operatorname{even}}$ is a smooth manifold with boundary, with bdf $r^{2}$. Throughout we shall denote the square root of our bdf $\rho=r^{2}$.

Now we state our main results on the behavior of solutions to the wave equation for small times. This question can be approached by a study of the fundamental solution to the wave equation, as in the work of Joshi-Sá Barreto [JoSá01] who studied the wave operator
$\cos\big{(}t\sqrt{\Delta_{g}-(n+1)^{2}/4}\big{)}$ in the setting of real asymptotically hyperbolic manifolds. This operator has Schwartz kernel $U(t,p,p^{\prime})$ satisfying

and they prove that $\cos\big{(}t\sqrt{\Delta_{g}-(n+1)^{2}/4}\big{)}$ resides in an algebra of Fourier integral operators. Having shown this, they use the results of [DuGu75, Hö68, Hör71] to study its (regularized) trace.

This construction of the wave group $U(t,p,p^{\prime})$ as a Fourier integral operator was motivated by the analysis of the resolvent of a real asymptotically hyperbolic manifold initiated in [MaMe87]. Mazzeo-Melrose obtained their results by exhibiting the resolvent as an element of the “large” calculus of zero pseudodifferential operators $\Psi_{0}^{*}(M)$; i.e., those pseudodifferential operators with Schwartz kernels constructed as distributions on the blown-up space $\overline{M}\times_{0}\overline{M}$, obtained by blowing up the intersection of the the corner $\partial\overline{M}\times\partial\overline{M}$ with the diagonal $\overline{M}_{\operatorname{diag}}\hookrightarrow\overline{M}\times\overline{M}$ in $\overline{M}\times\overline{M}$. The new boundary hypersurface resulting from this blow-up is called the front face. (For an extended treatment on such blow-ups see [MaMe87, §3], [Mel96], and [Gr01])

Along such lines [JoSá01] construct a class of zero Fourier integral operators as those operators whose Schwartz kernels, when lifted to $\overline{M}\times_{0}\overline{M}$, have support away from the left and right boundary faces (i.e. the lifts of $\partial\overline{M}\times\overline{M}$ and $\overline{M}\times\partial\overline{M}$ respectively). This greatly simplifies the construction of this class of operators, as typically the corners formed by the intersections of the left face (resp. right) with the front face would need to be incorporated into the definition of the operators; requiring the support of the Schwartz kernels avoid such corners allows their contributions to be neglected. In particular, due to the finite speed of propagation for the wave equation, a distribution which is initially supported only on the front face (such as $U(t,p,p^{\prime})$) will remain supported in the interior of the front face for all finite time. Thus [JoSá01] can construct a small time parametrix for the wave group while remaining entirely in this restricted calculus of zero Fourier integral operators.

Following this strategy we begin with the notion of the $\Theta$-stretched product, $\overline{X}\times_{\Theta}\overline{X}$, which is the analogous blow-up of the double space $\overline{X}\times\overline{X}$ defining the class of $\Theta$-pseudodifferential operators $\Psi_{\Theta}^{*}(X)$ used in the study of the resolvent initiated by [EpMeMe91]. With the appropriate definition of $\Theta$-Fourier integral operators, we can quickly conclude:

Once we know the wave group is a $\Theta$-Fourier integral operator, it is straightforward to use the results of [DuGu75, Hör71] to analyze the trace of $U(t,p,p^{\prime})$. One subtlety is that the trace needs to replaced with a regularized trace, defined using a Hadamard regularization procedure using our choice of bdf $\rho$. Defining the cut-off wave trace,

we obtain

With this result in hand, after choosing a smooth cutoff $\chi(t)\in{\mathcal{C}}^{\infty}_{0}(\mathbb{R})$ supported away from the lengths of all non-zero periods of closed geodesics, and using the results of [Hö68] we obtain a Duistermaat-Guillemin type result for the cutoff wave trace.

Finally, we remark on the appearance of the renormalized volume in Theorem (1.3). In the real hyperbolic setting it is known that the renormalized volume is, in certain dimensions, independent on the choice of representative of the conformal infinity. Namely, for $(M^{n+1},g)$ a real asymptotically hyperbolic manifold, one can similarly define the renormalized volume as the finite part of the in the expansion of $\operatorname{Vol}_{g}(\{x\geq\varepsilon\})$ as $\varepsilon\to 0$, given a choice of bdf $x$. For $n$ odd, the real hyperbolic renormalized volume is independent of $h_{0}$, the choice of conformal representative. On the other hand, for $n$ even, we suddenly have the dependence of the renormalized volume on this choice of representative of $[h]$. This is result is the so-called holographic anomaly (see [HeSk98]) and motivates much of the interest of asymptotically hyperbolic manifolds in mathematical physics, for their connection with the anti deSitter/conformal field theory (AdS/CFT) correspondence.

More concretely, the volume expansion of $(M^{n+1},g)$ of an Einstein asymptotically hyperbolic metric, for $n$ even, is given by

and [GrZw01] first made the connection of $V_{0}$ to Branson’s $Q$-curvature [Br95],

for $c_{n/2}$ a dimensional constant. In the ACH setting, the renormalized volume was first studied at this level of generality by Matsumoto in [Ma16]. Our construction of the renormalized wave trace thus provides an alternate proof of Matsumoto’s result, via formula (1.2). For a general ACH metric, [Ma16] generalizes this result for an analogue of Bransons $Q$-curvature. From his result we obtain as a corollary that the constant $d_{0}$ in our Theorem 1.3, is given by,

This quantity is a global CR invariant of the boundary, thus leading to a pseudoconformal analogue of the holographic anomaly. Given these results there is strong connection between the renormalized volume of an ACH manifold and its spectrum. On the mathematical physics side there seems to be relatively scarce work on this complex analogue of the AdS/CFT correspondence.

Funding This work was supported by the National Science Foundation grant numbers DGE-1746047, DMS-1440140, and DMS-1711325, growing out of conversations while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Fall 2019 semester.

Acknowledgements. The author is very grateful to acknowledge the kind discussions and suggestions of Andras Vasy, Antônio Sá Barreto, and Pierre Albin.

Because the construction of our adapted FIO-calculus entails a finer understanding of the geometry of an asymptotically complex hyperbolic manifold, we briefly recall the geometry of the Bergman-type metric our manifold is endowed with.

Let $(X,\partial X)$ be a non-compact manifold with closed boundary. We assume the boundary admits a contact form $\theta$ and an almost complex structure $J:\operatorname{Ker}(\theta)\to\operatorname{Ker}(\theta)$ (i.e., an endomorphism satisfying $J\circ J=-\text{Id}_{\operatorname{Ker}(\theta)}$) such that $d\theta(\cdot,J\cdot)$ is symmetric positive definite on $\operatorname{Ker}(\theta)$. We consider a metric $g_{ACH}$ of the following form: there is a boundary defining function $\rho$,

such that in a collar neighborhood $\varphi:[0,1)_{\rho}\times\partial X_{\omega,z}\to U$ it takes the form

where $(\mathcal{D}_{\rho}^{\mathbb{H}})^{*}Q_{\rho}$ is a smooth section of $S^{2}(T^{*}X)\cap\operatorname{Ker}(\iota_{\partial_{\rho}})$. Here, $\mathcal{D}_{\rho}^{\mathbb{H}}$ denotes the anisotropic dilation map

with splitting induced by the choice of contact structure $(\partial X,\theta)$, (i.e., $\mathscr{H}=\operatorname{Ker}\theta$).

We observe that for any other choice of defining function $\widetilde{\rho}$ we have

thus it is more natural to associate to $g_{ACH}$ a conformal class of 1-forms $[\Theta]$. The boundary manifold equipped with the data of $(\partial X,\theta,J)$ is a closed pseudohermitian manifold. The corresponding conformal pseudohermitian structure $([\Theta],J)$ was called a $\Theta$-structure in [EpMeMe91].

This Riemannian metric structure describes a non-compact incomplete manifold whose metric is asymptotic to complex hyperbolic space $\mathbb{H}_{\mathbb{C}}^{n+1}$. A useful model of complex hyperbolic space $\mathbb{H}_{\mathbb{C}}^{n+1}$ is given by

with boundary sphere equal to a compactification of the $(2n+1)$-dimensional Heisenberg group,

This model of complex hyperbolic space realizes $\mathbb{H}_{\mathbb{C}}^{n+1}\simeq\mathbb{R}^{+}\times H_{n}$ with the coordinates

foliating $\mathbb{H}_{\mathbb{C}}^{n+1}$ by a family of $H_{n}$-hypersurfaces. Writing $w=x+iy$, in these coordinates we can also write the contact form at the boundary as

and the metric on complex hyperbolic is the Bergman metric,

The Heisenberg group is a Lie group of dimension $2n+1$. In these coordinates the group law is given by

which is abelian in the first $2n$ components. Its Lie algebra $\mathfrak{h}$ has a basis $\{X_{j},Y_{j},Z\}$, which satisfies the non-trivial bracket relations: $[X_{j},Y_{j}]=Z$ for all $j=1,\ldots,n$ and all brackets vanishing. This structure of a nilpotent Lie algebra gives an identification $H_{n}\to\mathfrak{h}$ of the form

after which the group law can be written via Lie algebra elements $W,W^{\prime}\in\mathfrak{h}$ as,

It is a consequence the nilpotence of $H_{n}$ that the group law is a finite order polynomial in the Lie algebra elements, rather than the asymptotic series given by the Baker-Campbell-Hausdorff formula, (see e.g. [Ei68]).

Finally, we explain how the complexified hyperbolic space arises as a semi-direct product: there is parabolic dilation on $H_{n}$ (consistent with the bracket relations of the Lie algebra $\mathfrak{h}$) given by $\mathcal{D}_{\delta}(x,y,z)=(\delta x,\delta y,\delta^{2}z)$. The group law on the semidirect product $\mathbb{H}_{\mathbb{C}}^{n+1}\simeq\mathbb{R}^{+}\rtimes_{\mathcal{D}_{\delta}}H_{n}$ is given as

The geometric picture described above of complex hyperbolic space being foliated by a family of Heisenberg groups as level-set hypersurfaces of $\rho$ is compatible with this group law: an open set in $\{\rho=c\}\simeq H_{n}$ is related to the corresponding set in $\{\rho=c+\varepsilon\}$ by pullback along $M_{\varepsilon}$.

Our reason for expressing the Lie group law of $\mathbb{H}_{\mathbb{C}}^{m+1}$ at the level of its Lie algebra is that the Lie algebra arises more naturally at the level of tangent spaces in our later analysis.

In this section we begin the construction of a Fourier Integral Operator Calculus, which is adapted to the asymptotic geometry of the metric (2.1). Such a calculus will be comprised of operators whose Schwartz kernels have prescribed asymptotics on a manifold with corners, the $\Theta$-stretched product $X\times_{\Theta}X$ of [EpMeMe91].

Analogously to the 0-blow up, Epstein-Mendoza-Melrose defined the $\Theta$-blow up of an ACH manifold; this will be very similar to the zero-blow up of an AH manifold. The biggest distinction being the blow-up at the front face is non-isotropic, reflecting the different asymptotics in $\rho$ of boundary vector fields (namely those vector fields whose $g_{ACH}$-duals span $d\theta(\cdot,J\cdot)$ vs $\theta\otimes\theta$).

Following [EpMeMe91], we next explain how we will modify the product $X\times X$ to construct our algebra of Fourier integral operators. We begin with the notion of the $\Theta$-vector fields $\mathcal{V}_{\Theta}$:

where $\widetilde{\theta}\in{\mathcal{C}}^{\infty}(X;TX)$ is any smooth extension of $\theta$ to all of $X$. It is shown in [EpMeMe91, §1] that this definition is dependent only on the choice of conformal class of $[\theta]$. This is partly because a representative of $[\theta]$ determines a local frame by requiring

in which we can express

and a different choice of bdf $\rho^{\prime}$ produces a frame as in (3.1) associated to a contact form $\theta^{\prime}$ conformal to $\theta$.

Given this ${\mathcal{C}}^{\infty}(X)$-module, we can define the $\Theta$-tangent bundle ${}^{\Theta}TX$. This is a vector bundle over $X$, with a bundle map $\iota_{\Theta}:{}^{\Theta}TX\to TX$, which is an isomorphism over $X\setminus\partial X$ such that

Next, we construct the $\Theta$-stretched product of [EpMeMe91, §8]. Notice first that in the product $X\times X$, the boundary of the diagonal $\partial X_{\operatorname{diag}}\simeq\partial X$ is an embedded submanifold,

and is a clean submanifold in the sense of [DuGu75], since it is an embedded submanifold of the corner, and thus all differentials of bdfs vanish at $\partial X_{\operatorname{diag}}$. The 1-form $\theta$ on $X$ defines a line subbundle

spanned by

with $\pi_{(\cdot)}:X\times X\to X$ denoting the projection onto the left and right factors respectively.

With this trivialization of the conormal bundle, we define the $\Theta$-blow up of the corner as the $\mathscr{H}^{*}$-parabolic blow-up (defined using the dilation structure on fibers given in (2.2)) of the boundary diagonal:

where the equivalence on fibers $\mathcal{D}^{\mathscr{H}}$ is defined using the decomposition $N(\partial X_{\operatorname{diag}})_{+}=\mathscr{H}\oplus\mathscr{H}^{\perp},$ with $\mathscr{H}=\text{Ann}(\mathscr{H}^{*})$,

This real unoriented blow-up replaces the submanifold $\partial X_{\operatorname{diag}}$ with its inward-pointing parabolic-sphere bundle. This blow-up procedure furnishes a blow-down map

which is the identity on $X\times X\setminus\partial X_{\operatorname{diag}}$, and given by the bundle projection map of the parabolic-sphere bundle on $\mathbb{S}N_{\mathscr{H},+}(\partial X_{\operatorname{diag}})$. This is a manifold with corners, and has three new boundary faces:

By construction, the front face $\mathfrak{B}_{F}$ is a fiber bundle over $\partial X_{\operatorname{diag}}$ with fiber a projective quotient of the inward pointing normal bundle $N_{X\times X}(\partial X_{\operatorname{diag}})_{+}$; the front face has fiber over $p\in\partial X$ given by

For more details and the proof of diffeomorphism invariance of this construction see [EpMeMe91, §5-7].