Einstein metrics on strictly pseudoconvex domains from the viewpoint of bulk-boundary correspondence
aa r X i v : . [ m a t h . DG ] M a r EINSTEIN METRICS ON STRICTLY PSEUDOCONVEX DOMAINS FROMTHE VIEWPOINT OF BULK-BOUNDARY CORRESPONDENCE
YOSHIHIKO MATSUMOTO Introduction
This is a survey discussing some aspects of the correspondence proposed by Biquard [5, 6](see also [7]) between Einstein metrics on the interior of a manifold-with-boundary and strictlypseudoconvex CR structures on the boundary. Here, by “CR structures,” we shall mean not onlyintegrable almost CR structures but also certain nonintegrable ones, which will be described laterin this section.We may regard this correspondence as a differential-geometric interpretation and generaliza-tion of the classical complex-analytic correspondence between strictly pseudoconvex domains andCR manifolds arising as their boundaries. Let us start with describing this viewpoint.Fefferman’s mapping theorem [19] states that any biholomorphism Ω → Ω between smoothlybounded strictly pseudoconvex domains in C n ( n ≥
2) extends to a diffeomorphism between theclosures of the domains. This extension automatically restricts to a CR-diffeomorphism from ∂ Ω to ∂ Ω . Conversely, any CR-diffeomorphism ∂ Ω → ∂ Ω necessarily extends to a biholomorphismΩ → Ω by the Bochner–Hartogs theorem [11]. Such phenomena for domains in C n generalizeto those in Stein manifolds by the works of Bedford–Bell–Catlin [3] and Kohn–Rossi [35], and asa consequence, if D denotes the set of all smoothly bounded strictly pseudoconvex domains inStein manifolds, then biholomorphism classes of domains in D and CR-diffeomorphism classesof the boundaries of domains in D are in a one-to-one correspondence:(1.1) D / ∼ bihol ∼ = { ∂ Ω | Ω ∈ D } / ∼ CR-diffeo . The classical approach toward the correspondence (1.1) from differential geometry uses theBergman metric. However, here we would rather make use of the Einstein metric of Cheng–Yau[17] in the following theorem, because the Einstein equation has an advantage that it makessense without complex structures (recall that we are going to include some nonintegrable CRstructures on the boundary into our consideration).
Theorem 1.1 (Cheng–Yau [17]) . Let Ω be a smoothly bounded strictly pseudoconvex domain ina Stein manifold of dimension n ≥ . Then there exists a complete K¨ahler-Einstein metric withnegative Einstein constant on Ω , which is unique up to homothety. The metric is determined by the complex structure of Ω. On the other hand, the asymptoticbehavior of the metric at the boundary is mostly determined by the local CR geometry of ∂ Ω,as discussed by Fefferman [24] and Graham [26]. In this sense, the Cheng–Yau metric is alink that realizes (to some extent) the correspondence (1.1). We will introduce the notion of“asymptotically complex hyperbolic Einstein metrics” in the next section, for which the Cheng–Yau metrics serve as model examples.Now we illustrate the class of almost CR structures that we consider. Let M be a (connected)differentiable manifold of dimension 2 n −
1, where n ≥
2, and H a contact distribution over M . We say, in this article, that an almost CR structure J on H (i.e., J ∈ Γ(End( H )) satisfying J = − id) is compatible when the Levi form h θ ( X, Y ) := dθ ( X, JY ) , X, Y ∈ H is symmetric in X and Y for some (hence any) contact 1-form θ annihilating the distribution H .(If H ⊥ ⊂ T ∗ M is oriented, then H has a natural CSp ( n − H from CSp ( n − CU ( p, n − − p ) for some p . The term “compatible” refers to the compatibility of J to the CSp ( n − H in this setting.) Integrable almost CR structures are always compatibleas is well known, but there are more compatible structures (except for the three-dimensional case,in which any almost CR structure is automatically integrable). It can be easily checked that J is compatible if and only if(1.2) [Γ( H , J ) , Γ( H , J )] ⊂ Γ( H C ) , where H C is the complexification of H and H C = H , J ⊕ H , J is the eigenbundle decompositionwith respect to J (note that (1.2) is not a trivial condition since H C ( T C ∂ Ω). Because of(1.2), compatible almost CR structures are also called partially integrable in the literature (e.g.,[13, 14, 43, 45, 44]).The usual notion of strict pseudoconvexity naturally extends to compatible almost CR struc-tures. Namely, a compatible almost CR structure J is said to be strictly pseudoconvex if the Leviform h θ has definite signature. In the following we shall always assume the strict pseudoconvexity,and a contact form θ is always taken so that h θ is positive definite.Each asymptotically complex hyperbolic (ACH for short) Einstein metric is “associated” to,or “fills inside” of, a manifold equipped with a strictly pseudoconvex compatible almost CRstructure, as the Cheng–Yau metric does. Relationships between Einstein metrics and geometricstructures on the boundary have been more actively studied in the setting of Poincar´e-Einstein(or AH-Einstein) metrics and conformal structures, partly because of physical interests. Fur-thermore, the cases of Poincar´e-Einstein and ACH-Einstein metrics are generalized to a broaderperspective involving “asymptotically symmetric Einstein metrics” and “parabolic geometries,”which is illustrated in [5, 6, 9]. The term “bulk-boundary correspondence” in the title of thisarticle is intended to indicate this very general correspondence, most part of which is yet to beunveiled. 2. Asymptotically complex hyperbolic Einstein metrics
In order to motivate our definition of ACH metrics, let us first observe the fact that the leadingpart of the asymptotic behavior of the Cheng–Yau metric g at the boundary can be described interms of the CR structure of ∂ Ω.From the proof of its existence, it is known that g is expressed (after a normalization) as(2.1) g ij = ∂ i ∂ j log 1 ϕ = ( ∂ i ϕ )( ∂ j ϕ ) ϕ − ∂ i ∂ j ϕϕ , where ϕ ∈ C ∞ (Ω) ∩ C n +1 ,α (Ω) is some defining function of Ω, i.e., Ω = { ϕ > } and dϕ isnowhere vanishing on ∂ Ω, where α ∈ (0 ,
1) is arbitrary (to be precise, [38] is responsible for thisoptimal boundary regularity). Because of (2.1), one can take a diffeomorphism of the form(2.2) Φ = ( π, ρ ) :
U → ∂ Ω × [0 , ε ) , INSTEIN METRICS ON STRICTLY PSEUDOCONVEX DOMAINS 3 where U is an open neighborhood of ∂ Ω in Ω, such that π : U → ∂ Ω restricts to the identity mapon ∂ Ω and the Cheng–Yau metric g satisfies(2.3) g ∼ Φ ∗ g θ , g θ = 12 (cid:18) dρ ρ + θ ρ + h θ ρ (cid:19) as ρ →
0, where θ is some contact form on ∂ Ω annihilating the natural contact distributionand h θ is the associated Levi form. The meaning of (2.3) can be understood as, for example,that | g − Φ ∗ g θ | Φ ∗ g θ uniformly tends to 0 as ρ →
0. More is true actually: it follows from theasymptotic expansion established in [38] that the C k norm of ρ − ( g − Φ ∗ g θ ), defined geometricallyby Φ ∗ g θ , is finite for any k ≥
0. We note that there exists, for any choice of θ , a diffeomorphismΦ with respect to which (2.3) holds—there is no preferred choice of θ . We also remark that inthe literature the model metric g θ is sometimes expressed as g θ = 12 (cid:18) dx x + θ x + h θ x (cid:19) by introducing a new coordinate x = √ ρ .Observing the asymptotic behavior (2.3) of the Cheng–Yau metric, we define as follows. Met-rics of this type are firstly considered by Epstein–Melrose–Mendoza [18], in which the meromor-phic continuation of the resolvent of the Laplacian (on functions) is studied. Definition 2.1.
Let X be a compact smooth manifold-with-boundary with dim R X = 2 n , where n ≥
2, and X its interior. A Riemannian metric g defined on X is called an asymptotically complexhyperbolic metric (or ACH metric ) when there exists a diffeomorphism Φ like (2.2) such that g satisfies (2.3) with respect to some contact distribution H over ∂X , a strictly pseudoconvexcompatible almost CR structure J on H , and a contact form θ in the sense that g − Φ ∗ g θ ∈ C ,αδ ( X, S T ∗ X )for some δ > α ∈ (0 , C k,αδ ( X, S T ∗ X ) denotes the space of C k symmetric2-tensors σ on X such that ρ − δ/ σ has finite C k,α norm with respect to Φ ∗ g θ (this space dependson Φ and H , but not on J ). The almost CR structure J , or the triple ( ∂X, H, J ), is called the conformal infinity of g .Our fundamental questions on ACH metrics are the following. For a given X , does there existan Einstein ACH metric on X with prescribed conformal infinity? If there does, how many arethere essentially (i.e., up to the action of diffeomorphisms)?Let us focus on the existence problem for the moment. The Cheng–Yau theorem (Theorem1.1) provides many examples of Einstein ACH metrics, but for general infinity, only perturbativeresults are known. Such results are given by Roth [46], Biquard [6], and the present author [44],which we shall now discuss.In [46] and [6], general perturbation theory is established. Roth considered deformations ofthe Cheng–Yau metrics, while Biquard worked on those of arbitrary Einstein ACH metrics. Itwas shown that, in the both works, that if the given Einstein metric g has negative sectionalcurvature everywhere, then compatible almost CR structures nearby the conformal infinity of g are also “fillable” with Einstein metrics. More precisely, the following theorem holds. Theorem 2.2 (Biquard [6]) . Let g be an Einstein ACH metric on X , whose conformal infinityis denoted by ( ∂X, H, J ) . Suppose that g has negative sectional curvature. Then, if J is asufficiently small C ,α neighborhood of J in the space of compatible almost CR structures on H ,any J ∈ J is the conformal infinity of some Einstein ACH metric on X . YOSHIHIKO MATSUMOTO
In particular, Theorem 2.2 is applicable to the complex hyperbolic metric on the unit ball B n in C n (note also that it is the Cheng–Yau metric of B n ). Leaving the contact distribution H unchanged is not an additional restriction, because contact structures of closed manifolds arerigid.Here is a very brief sketch of the construction (which is discussed more in the next section).We first assign to each J ∈ J an approximate ACH solution g J of the Einstein equation whichsatisfies Ric( g J ) + ( n + 1) g J ∈ C ,αδ ( X, S T ∗ X )for some δ > δ must be independent of J ). We can do it in such a way that g J is smooth in J and g J equals the original metric g . Then we use functional analysis to show that, making J smaller if necessary, for each J ∈ J one can find σ ∈ C ,αδ ( X, S T ∗ X ) for which g ′ J = g J + σ satisfies Ric( g ′ J ) = − ( n + 1) g ′ J . Since the modification term σ belongs to C ,αδ , g ′ J is still an ACHmetric whose conformal infinity is J .The negative curvature assumption is an easy sufficient condition that makes this plan work.However, in practice, it is a nontrivial matter to check whether this condition is satisfied fora given g . The following theorem shows that it is unnecessary for the Cheng–Yau metrics, (atleast) except for the two-dimensional case. Theorem 2.3 (Matsumoto [44]) . Let Ω be a smoothly bounded strictly pseudoconvex domainin a Stein manifold of dimension n ≥ , and J a sufficiently small C ,α neighborhood of theinduced CR structure J in the space of compatible almost CR structures on the natural contactdistribution over ∂ Ω . Then for each J ∈ J , there is an Einstein ACH metric on Ω with conformalinfinity J . There are such perturbation theorems also for Poincar´e-Einstein metrics. The possibility ofdeforming the real hyperbolic metric is shown by Graham–Lee [28], and in [6] it was pointed outthat the negative curvature assumption is sufficient. Lee [37] showed that a weaker curvatureassumption suffices when the boundary conformal structure has nonnegative Yamabe constant.In [6], the local uniqueness of Einstein ACH metrics is also discussed. By shrinking J ifnecessary, the Einstein metric g ′ J constructed for each J ∈ J is the unique Einstein metric modulodiffeomorphism action in a neighborhood of g ′ J in g ′ J + C ,αδ ( X, S T ∗ X ) (for any δ > g in the unweighted H¨olderspace C ,α ( X, S T ∗ X ) in which there is only one Einstein metric for each conformal infinity?To the author’s knowledge, this is not yet settled so far.3. Ideas of the Proofs of Theorems 2.2 and 2.3
The two theorems in the previous section are reduced to the vanishing of the L kernel of the“linearized gauged Einstein operator” acting on symmetric 2-tensors, which is(3.1) P = 12 ( ∇ ∗ ∇ − R ) , where g is the given Einstein ACH metric and ˚ R denotes the pointwise linear action of thecurvature tensor of g . Let us see how this reduction is carried out.It is natural to study the linearization of the Einstein equation in order to deform Einsteinmetrics. However, if we consider the Einstein equation itself, we encounter a difficulty that orig-inates from the diffeomorphism invariance of the equation. One usually introduces an additional INSTEIN METRICS ON STRICTLY PSEUDOCONVEX DOMAINS 5 term to break this “gauge invariance.” Here we set, following [6], E g ( g ′ ) := Ric( g ′ ) + ( n + 1) g ′ + δ ∗ g ′ B g ( g ′ ) , B g ( g ′ ) := δ g g ′ + 12 d tr g g ′ . As long as we consider g ′ in a small neighborhood of g in g + C ,αδ ( X, S T ∗ X ), any solution of E g ( g ′ ) = 0 automatically satisfies B g ( g ′ ) = 0, and hence it becomes an Einstein metric.We apply the implicit function theorem to the mapping J × C ,αδ ( X, S T ∗ X ) → C ,αδ ( X, S T ∗ X ) , ( J, σ )
7→ E g J ( g J + σ )at ( J , g J is a family of approximate solutions as described in the previous section. Ifthe linearization of σ
7→ E g ( g + σ ) at σ = 0, which is the operator (3.1), is invertible, then foreach J ∈ J sufficiently close to J there exists σ ∈ C ,αδ ( X, S T ∗ X ) satisfying E g J ( g J + σ ) = 0.Thus it suffices to prove that (3.1) is an isomorphism as the mapping(3.2) P : C ,αδ ( X, S T ∗ X ) → C ,αδ ( X, S T ∗ X )for sufficiently small δ > δ > P : H ( X, S T ∗ X ) → L ( X, S T ∗ X )is isomorphic, where H ( X, S T ∗ X ) denotes the L Sobolev space of order 2, which is actuallythe domain of P seen as an unbounded operator on L ( X, S T ∗ X ). It is easy to show that P isa self-adjoint unbounded operator, and hence (3.3) is isomorphic if the L kernel vanishes. Theequivalence of (3.2) and (3.3) being isomorphic follows by a certain parametrix construction,which makes good use of the geometry of ACH metrics, explained in [6]. The exposition on thePoincar´e-Einstein case in [37] is also useful.Consequently, it suffices to show that the L kernel of P is trivial. When g has negativesectional curvature, the vanishing can be proved by the following Bochner technique. Note thatany element of the kernel must be trace-free, because if σ = ug for some u ∈ C ∞ ( X ) then P σ = ( ∇ ∗ ∇ u + 2( n + 1) u ) g , and the operator ∇ ∗ ∇ + 2( n + 1) acting on functions has trivial L kernel. Now if a general symmetric 2-tensor σ is regarded as a 1-form with values in T ∗ X , then P satisfies the Weitzenb¨ock formula below given in terms of the exterior covariant differentiation D (see [4, 12.69]): 2 P σ = ( DD ∗ + D ∗ D ) σ − ˚ Rσ + ( n + 1) σ. Moreover, there is also a pointwise estimate valid for trace-free σ ([4, 12.71]) that h ˚ Rσ, σ i ≤ ( n + 1 + 2( n − K max ) | σ | , where K max is the maximum of the sectional curvatures at a point. Since the assumption impliesthat the sectional curvature is bounded from above by a negative constant (by virtue of theasymptotic complex hyperbolicity), one can deduce that the L kernel of P is trivial in this case.When g is the Cheng–Yau metric of a smoothly bounded strictly pseudoconvex domain Ω in aStein manifold, we argue as follows based on Koiso’s observations [36] (see also Besse [4, Section12.J]). The K¨ahlerness of g implies that P respects the type decomposition of σ into hermitianand anti-hermitian parts, and due to the Einstein condition, P on each type becomes a familiaroperator. If σ is hermitian, then one may regard it as a (1 , P σ = ( dd ∗ + d ∗ d ) σ + 2( n + 1) σ. YOSHIHIKO MATSUMOTO
This shows the vanishing of the hermitian part of the L kernel. On anti-hermitian symmetric 2-tensors, by regarding them as (0 , T , Ω,we obtain 2
P σ = ( ∂ ∂ ∗ + ∂ ∗ ∂ ) σ. Therefore it suffices to show that there are no nontrivial L harmonic (0 , T , Ω. This allows one to restate the problem in terms of cohomology: the isomorphicity of(3.2) for small δ > L Dolbeault cohomology H , (Ω , T , Ω) vanishes.It is a consequence of classical theory on Stein manifolds that the compactly supported co-homology H , c (Ω , T , Ω) vanishes. Moreover, it can be observed that the following sequenceinvolving the inductive limit lim −→ K H , (Ω \ K ; T , (Ω \ K )), where K runs through all compactsubsets of Ω, is exact: · · · → H , c (Ω; T , Ω) → H , (Ω; T , Ω) → lim −→ K H , (Ω \ K ; T , (Ω \ K )) → H , c (Ω; T , Ω) → · · · . Therefore, H , (Ω , T , Ω) = 0 follows if(3.4) lim −→ K H , (Ω \ K, T , (Ω \ K )) = 0holds. We show (3.4) by proving(3.5) H , ( U , T , U ) = 0 , where U is a sufficiently narrow collar neighborhood of ∂ Ω intersected with Ω. The vanishing(3.5) is attacked by the usual technique of L estimate, but one needs to be careful becauseboundary integrals along the inner boundary of U , which is strictly pseudoconcave, comes intoplay. The L estimate so obtained is actually sufficient to prove (3.5) only when n ≥
4. When n = 3, one needs to work with a weighted L cohomology instead.4. Problems
An obvious problem related to Theorem 2.3 is to clarify what happens in the two-dimensionalcase. The author expects (perhaps optimistically) that finally one can simply remove the as-sumption n ≥ nonexistence result by Gursky–Han [29] in the latter setting.Turning to the uniqueness of Einstein fillings for a given conformal infinity, in the Poincar´e-Einstein case, an example of Hawking–Page [30] exhibits that it fails in general (see Anderson[1] for further explanation). A similar nonuniqueness example for ACH-Einstein metrics will beof great interest, as well as uniqueness results under some assumption. There is also a room forfurther investigations about local uniqueness as mentioned at the end of Section 2.A typical application of Poincar´e-Einstein metrics is the construction of conformally invari-ant objects on the boundary, and there is a similar story for ACH-Einstein metrics. For thispurpose the determination of the asymptotic behavior of the metric in terms of the boundarygeometry is important, and its formal aspects are studied by Fefferman–Graham [21, 22] for thePoincar´e-Einstein metrics, by Fefferman [24] and Graham [26] for the Cheng–Yau metrics (asmentioned in Section 1), and by Biquard–Herzlich [8] and the author [43] (see also [42]) for gen-eral ACH-Einstein metrics. There is a tremendous amount of literature regarding constructions INSTEIN METRICS ON STRICTLY PSEUDOCONVEX DOMAINS 7 of conformal invariants based on [21, 22], while in the CR case such constructions are discussedin, e.g., [20, 26, 27, 12, 2, 31, 23, 8, 25, 34, 16, 32, 39, 45, 33, 47, 15, 41, 48, 49, 40]. Further develop-ments along this line are anticipated. It would be also very interesting if there is some invariantconstruction that needs global considerations on Einstein metrics in an essential way.Now let us take notice of the fact that the Cheng–Yau metrics come with complex structureswith respect to which they are K¨ahler. As a problem without any counterpart in the Poincar´e-Einstein setting, it may be interesting to look for a canonical way to determine a good almostcomplex structure on a manifold equipped with an ACH-Einstein metric. That is to say, Rie-mannian metrics may not be the “best” filling geometric structure inside CR manifolds. It seemsto the author that this idea is backed up by the fact that the Einstein deformation problem is re-cast in the proof of Theorem 2.3 in terms of harmonic (0 , Acknowledgments
I wish to express my gratitude to the hospitality of Stanford University, where I was work-ing as a visiting member when the manuscript was written, and I am deeply grateful to RafeMazzeo for hosting the visit, for discussions, and for encouragements. I would also appreciatethe careful reading of the manuscript by the reviewer. This work was partially supported byJSPS KAKENHI Grant Number JP17K14189 and JSPS Overseas Research Fellowship.
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Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka560-0043, JapanDepartment of Mathematics, Stanford University, Stanford, CA 94305-2125, USA
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