A remark on renormalized volume and Euler characteristic for ache 4-manifolds

Differential Geometry and its Applications 25 (2007) 78–91 www.elsevier.com/locate/difgeo

A remark on renormalized volume and Euler characteristic for ACHE 4-manifolds Marc Herzlich Institut de Mathématiques et Modélisation de Montpellier, UMR 5149 CNRS, Université Montpellier II, Place E. Bataillon, 34095 Montpellier Cedex 5, France Received 23 April 2004; received in revised form 7 October 2005 Available online 6 June 2006 Communicated by J. Slovak

Abstract This note computes a renormalized volume and a renormalized Gauss–Bonnet–Chern formula for asymptotically complex hyperbolic Einstein (so-called ACHE) 4-manifolds. © 2006 Elsevier B.V. All rights reserved. MSC: 53C20; 53C25; 53C55 Keywords: Asymptotically complex hyperbolic Einstein metrics; Euler characteristic; Renormalized volume

1. Introduction Asymptotically symmetric Einstein metrics exhibit many interesting phenomena [4,13]. They were especially studied in the Einstein asymptotically hyperbolic (or AHE) case, which enjoys fruitful relationships with physics through the ADS - CFT correspondence, but also is a useful tool establishing links between the conformal geometry of a compact (n − 1)-dimensional manifold (usually called the boundary at infinity) and the Riemannian geometry of a complete Einstein n-dimensional manifold (the bulk AHE manifold). In this setting, an intriguing invariant, called renormalized volume, has been defined by C.R. Graham [12], after works by physicists such as Henningson and Skenderis [14]. In even dimensions n, the renormalized volume is an invariant of the AHE metric. Its role in the formula for the Euler characteristic of the Einstein manifold has been moreover pointed out by M.T. Anderson [2] in dimension 4, by S.-Y.A. Chang, J. Qing and P. Yang [7] and by P. Albin [1] in higher dimensions, with applications in dimension 4 to the study of the moduli space of Einstein asymptotically real hyperbolic metrics [3]. This is a “renormalized Gauss– Bonnet–Chern formula” since the Einstein manifold is non-compact, but all divergent terms in the integrals of the formula are shown to cancel, whereas renormalized volume appears as a finite limit contribution. In odd dimensions n, the renormalized volume is not an invariant of the AHE metric only but depends on a choice of a representative metric on the boundary at infinity in its conformal class. This makes it no less interesting, as it

E-mail address: [email protected] (M. Herzlich). 0926-2245/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.difgeo.2006.05.005

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gives rise to the so-called conformal anomaly phenomenon: the difference between the renormalized volumes of two different choices of metric singles out a local differential operator on the boundary with nice properties [12]. The goal of this short note is to point out analogous results in the case of Einstein asymptotically complex hyperbolic (or ACHE) manifolds of dimension 4, where the boundary at infinity is now a strictly pseudoconvex CR 3-manifold, with the hope that such an object would be interesting for the study of 3-dimensional CR geometry. Starting with an Einstein asymptotically complex hyperbolic (ACHE) metric g on a 4-dimensional M with boundary at infinity a CR manifold X, it has been shown in [5] that there exists a (formal) Kähler–Einstein metric g¯ on a neighbourhood of infinity on M whose first terms are formally determined by the choice of a compatible contact form (or in the usual language of CR geometry: a pseudo-hermitian structure) on X. One can then show that it is always possible to find a diffeomorphism ψ inducing the identity at infinity so that ψ ∗ g admits a nice asymptotic development whose first terms are given by the metric g. ¯ The volume form of ψ ∗ g itself has an asymptotic development, and we then define the renormalized volume as the term of order 0 in this expansion. This is well defined, as we can prove: Theorem 1.1. Let (M, g) be a 4-dimensional Einstein asymptotically complex hyperbolic (ACHE) manifold, with boundary at infinity a compact strictly pseudoconvex CR 3-dimensional manifold X. For any choice of compatible contact form η on X, let g¯ be the associated formal Kähler–Einstein metric in a neighbourhood of infinity in M, and ψ a diffeomorphism as above. Then the renormalized volume V is well defined and only depends on g and the choice of η on X; we shall call it the renormalized volume of (M, g) relative to η. A detailed analysis of the boundary term of the expression of the Euler characteristic of M in terms of the curvature of g shows that a renormalized Gauss–Bonnet characteristic formula can be obtained. The result reads as follows: Theorem 1.2. Let (M, g) be a 4-dimensional Einstein asymptotically complex hyperbolic (ACHE) manifold, with boundary at infinity a compact strictly pseudoconvex CR 3-dimensional manifold X. For any choice of compatible contact form η leading to a renormalized volume V , one has     2 (Scalg )2 1 3 5 2 R 1 g 2 |W | − V − = − |τ | η ∧ dη, χ(M) − (1.1) 24 16 2 8π 2 8π 2 4π 2 M

X

where R and τ are the curvature and torsion of the Webster–Tanaka connection determined by η. Note that it is shown in [5] that the integrals of both terms 1 Scal2 24 converge on an ACHE manifold, so that the formula above makes sense. By applying the previous theorem, we get another proof of the fact that the renormalized volume depends only on (M, g) and η, since it explicitly relates V with objects defined only in terms of g and a choice of contact form at infinity. Moreover, this implies of course that the number    2 5 R 3 − |τ |2 η ∧ dη V= V− (1.2) 2 16 2 |W − |2

and |W + |2 −

X

is an invariant of the ACHE manifold (M, g) only. The situation depicted in Theorem 1.2 is then less pleasant than in the AHE case, as the renormalized volume is never an invariant of the complete Einstein metric and always depends on the choice of a contact form at infinity. As the model case of the complex hyperbolic plane shows, the appearance of a local correction on the boundary seems unavoidable. This situation is of course reminiscent from the AHE case of odd dimension (i.e. boundary at infinity of even dimension); this should come as no surprise as it is usually considered that CR geometry enjoys lots of analogies with even-dimensional conformal geometry. It moreover shows that, rather than giving rise to a global invariant, the renormalized volume gives birth to a conformal (or rather CR) anomaly, i.e. a formula relating the renormalized

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volume for some choice of pseudo-hermitian structure at infinity to its expression for some other choice at infinity, through a local differential expression. Namely, Corollary 1.3. For each contact form η, there is a differential operator Pη on X such that, for any function f on X which never vanishes,  V (f η) − V (η) = Pη (f ) η ∧ dη. X

In the AHE case [12], the conformal anomaly is given by differential operators with nice properties. Our result 2 suggests that it could be interesting to study the operator arising from the variation of R16 − 52 |τ |2 under deformations of the contact form in the same contact structure. The operator Pη can be easily computed by using the transformation laws of R 2 and |τ |2 to be found in [16,17]. Note however that the CR anomaly in the renormalized volume is in a sense harmless since it is enough to add a local contribution at infinity to recover an invariant. After this paper had first appeared in the ArXiv preprint database, N. Seshadri has given in [18] another version of the renormalized volume that covers all dimensions but for Kähler–Einstein metrics only rather than ACHE metrics. A comparison between both formulas is done in Section 4. 2. Definitions and notations Let (X 3 , H, J0 ) be a strictly pseudo-convex 3-dimensional CR manifold, i.e. a contact manifold with contact distribution H and almost complex structure J0 on H . If η in any choice of compatible contact form, an associated metric γ may be defined on H by γ = dη(·, J0 ·). The Reeb field is the vector field ξ defined by η(ξ ) = 1 and ιξ dη = 0.

(2.1)

To each choice of η is attached a compatible connection ∇ called the Tanaka–Webster connection; its torsion in the direction of ξ will be denoted by τ = ∇ξ · −[ξ, ·]. Let M be a 4-manifold such that the complement of some compact set is diffeomorphic to [r0 , +∞[ × X. We consider first the metric g0 = dr 2 + e2r η2 + er γ on ]r0 , +∞[ × X and let Cδ∞ be the space of smooth functions or tensors on M such that eδr ∇ k f is bounded for any k. Any metric g on M such that g − (dr 2 + e2r η2 + er γ ) belongs to Cδ∞ for some δ > 0 will be called an asymptotically complex hyperbolic (ACH) metric. Moreover, (M, g) is said to be Einstein asymptotically complex hyperbolic, or ACHE, if g is an Einstein metric. A lot of such metrics arise on pseudoconvex domains in C2 (and are Kähler–Einstein in this case [9]) whereas another important family was constructed by O. Biquard in [4]. The Biquard metrics are especially interesting in the case the boundary at infinity X is endowed with a nonembeddable CR structure, as they provide a substitute for the globally non-existing Kähler–Einstein metric. In [5], the author and O. Biquard carefully obtained precise asymptotic expansions of ACHE metrics in dimension 4. They are crucial for the present work, and we shall now describe them in some detail. In all that follows, we consider an ACHE metric g on a neighbourhood of infinity ]r0 , +∞[ × X in M. If a contact d... ¯ will denote the components of S, and (a, b, c, d, . . . ∈ {0, 1, 1}) form η is given, then for any tensor field S on X, Sa,bc... subsequent Tanaka–Webster derivatives separated by a comma from the original components, in a local orthonormal ¯ ¯ a coframe (η, θ 1 , θ 1 ), such that dη = iθ 1 ∧ θ 1 . For instance we shall use expressions such as τb,c... , R,ab... for the (derivatives of the) torsion and curvature of the Tanaka–Webster connection. Last, in any power series expansion  φk (x) ekr , the kth term φk (seen as a function or section on X) will be called formally determined if it can be computed with the knowledge of a finite jet of the CR structure or contact form at x ∈ X only. The most interesting feature of ACHE metrics (and Kähler–Einstein metrics as well) is that they are not entirely formally determined. The results in [5] are described in the following statements: Theorem 2.1. [5] There exists on ]r0 , +∞[ × X an integrable complex structure J given by a (not necessarily convergent) power series, entirely determined formally from data at infinity, and given by: J ∂r = e−r ξ,

J ξ = er ∂r ,

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   5  J|H = J0 − 2 e−r τ + e−2r 2|τ |2 − J0 ∇ξ τ + O e− 2 r ,

81

(2.2)

or equivalently by the choice of a coframe (ϑ 0 = e−r dr + iη, ϑ 1 = θ 1 − φθ 1 ) of type (1, 0), where φ is a map to HJ1,0 whose first terms are from HJ0,1 0 0 φ = −i e−r τ +

 5  1 −2r e ∇ξ τ + O e− 2 r . 2

(2.3)

5

The decays O(e− 2 r ) in the previous statements are computed with respect to the metric g0 = dr 2 + e2r η2 + er γ . It is a fact that the same decay is true for all derivatives; we shall not repeat it unless it plays a special role in a proof. This useful convention will be in order for the rest of the paper.  Once this complex structure is obtained, one gets from [5, Corollaries 3.7 and 5.4] (expanding the expression of Ω that appears there): ¯ whose Kähler form ω is uniquely Theorem 2.2. [5] There is on ]r0 , +∞[ × X a (formal) Kähler–Einstein metric g, formally determined up to order 2 at least, up to J -biholomorphisms, as follows: R ω = er (dr ∧ η + dη) − dη 2   4 i i 1 1 0 1 1¯ 0¯ 0 1 1 1¯ 0¯ 1¯ + ϑ ∧ θ − τ1,1¯ ϑ ∧ θ R ¯ ϑ ∧ θ − R,1 ϑ ∧ θ − τ1,1 3 8 ,1 8 2 ¯ 2   2  −r R 2i  1 2 R 1¯ e dr ∧ η − τ1, − |τ |2 − + τ1,11 − ¯ ¯ 1¯ 1 3 8 6 3    −r  5  R i 1 2 R2 1¯ e dη + O e− 2 r . − τ − |τ |2 + − τ1,11 + ¯ ¯ 1¯ 1, 1 3 8 12 3 A little bit more of computation then gives an explicit asymptotic expansion for the metric g¯ associated with J and ω: Corollary 2.3. The Kähler metric g¯ is explicitly given by  R 1 ¯ ¯ γ + 2γ (J0 τ · ,·) + R,1 θ 1 ◦ ϑ 0 + R,1¯ θ 1 ◦ ϑ 0 2 6    2i  1 0 1¯ 1 −r −r 1¯ 0¯ + τ1,1 ¯ ϑ ◦ θ − τ1,1¯ ϑ ◦ θ − e Rγ (J0 τ · ,·) − e γ ∇ξ τ (·), · 3    −2r R 2i  1 2 R2 2 1¯ − |τ | − + τ ¯ − τ1,1¯ 1¯ e (dr 2 + e2r η2 ) − 3 8 6 3 1,11    −r  5  R i 1 2 R2 1¯ e γ + O e− 2 r , − τ − |τ |2 + − τ1,11 + ¯ ¯ 1¯ 1, 1 3 8 12 3

g¯ = (dr 2 + e2r η2 + er γ ) −

where α ◦ β = α ⊗ β + β ⊗ α is the symmetrized product of 1-forms. Following [5, Section 5] again, we see that for any ACHE metric g with the same boundary at infinity, there exists a smooth section k over X of the bundle of anti-J0 -invariant symmetric bilinear forms on H (seen here as a field of bilinear forms on T M by setting k(ξ, ·) = k(∂r , ·) = 0), a diffeomorphism ψ asymptotic to identity at infinity, and a positive real number δ such that ψ ∗ g − g¯ − k e−r = O(e−(2+δ)r ),

(2.4)

O(e−(2+δ)r )

estimate applies as before to all derivatives as well (and the reader is warned that the term k e−r where the −2r decays like e ). In a collar neighbourhood of infinity (i.e. [R1 , +∞[ × X with R1 large enough) the diffeomorphism ψ is close enough to identity to be written as ψ : p −→ ψ(p) = expp X(p)

(2.5)

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where eτ r X is bounded as well as all its derivatives for some τ > 0, and the exponential map refers to the fixed metric g0 (details on this point in the entirely analogous AH case have been given in [10]; note that this fact should have been underlined more clearly in [4,5]). An important point here is that there is no reason why the diffeomorphism ψ should be unique, at least up to terms of strictly faster decay than e−2r , that will play no role in what follows. The required uniqueness is however true and can be stated in the following form: Proposition 2.4. Let φ a diffeomorphism of the type φ : p −→ φ(p) = expp X(p)

(2.6)

with eτ r X bounded as well as all its derivatives for some τ > 0, such that φ ∗ g¯ = g¯ + κ e−r + h where e(2+δ)r h as well as all its derivatives are bounded for some δ > 0, and κ is a J0 -antilinear symmetric bilinear  form on H . Then e(2+δ )r X and all its derivatives are bounded in g0 -norm for some δ  > 0. This can be restated in a more formal way by saying that X belongs to some weighted functional space, with specific values of the weight. Namely, we define (for functions or sections of a bundle) Cρ∞ = {u ∈ C ∞ , eρr u ∈ C ∞ } ¯ Further properties of these spaces are studied in [4,5]. The where derivatives are taken with respect to either g0 or g. weighted spaces language will be useful at one precise step of the proof to follow. Otherwise we have tried to avoid such technicalities (e.g. in the statement of Proposition 2.4). Proof. Roughly speaking, the idea is to show that any such vector field cannot have any non-zero dominant term in its expansion, unless it is of very fast decay. More precisely, let φ be a diffeomorphism as in the statement of the proposition. Then g¯ + LX g¯ + Q(X, g) ¯ = g¯ + κ e−r + h ¯ as well as all its derivatives are bounded (from now on, whenever we say that some quantity eρr u where e2τ r Q(X, g) ¯ we get is bounded, this will include the same statement for all derivatives of u). Hence, if we let (δ g¯ )∗ X := LX g, (δ g¯ )∗ X = S1 ∈ Cρ∞1 ,

with ρ1 = min(2τ, 1 + τ, 2).

(2.7)

The usual Weitzenböck formula is δ g¯ (δ g¯ )∗ = g¯ + ∇ g¯ δ g¯ − Ricg¯ , where  is the rough Laplacian on vector fields. Since δ g¯ X = 12 tr(δ g¯ )∗ X = 12 tr S1 , and since g¯ can be taken to be Einstein at a very high order, we get that   3 g¯ + (2.8) X = S2 , 2 with S2 in the same weighted space as S1 . We then apply [4, §I.2]: Kato’s inequality ensures that the largest critical

weight of g¯ + 32 is at least 1 + 1 + 32 > 2 whereas the lowest is negative, so that X ∈ Cρ∞1 whenever ρ1  2. By an elementary bootstrap argument, we get that X ∈ C2∞ . We now come back to (2.8). Following the argument presented in [5, Proposition 5.2], we can show that tangential derivatives of X decay faster than expected. More precisely, we denote by ∇ W the natural extension of the Tanaka– Webster connection on tensor fields on [R0 , +∞[ × X, see [5, §2], and we shall prove that ∞ ∇ξW X ∈ C1+δ ,

δ

∇hW X ∈ C ∞ , 3 +δ  2

for some > 0 and any γ -unit local section h on an open set U ⊂ X extended to [R0 , +∞[ × U in the obvious way. Indeed, one may write (2.8) slightly more precisely as   3 g¯ + X = κ2 + T2 , 2

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where κ2 collects all terms issued from κ and its derivatives and T2 is the remainder. As a result, κ2 is a sum of e−ar terms with coefficients being smooth vector fields on the boundary at infinity, and any such term is of global decay e−2r at least. On the other hand, T2 belongs to Cρ∞2 for some ρ2 > 2. We now let P = (g¯ + 32 ) and take h to be any γ -unit local section of H on the boundary at infinity. Lemma 2.6 of [5] shows that the commutator [P , ∇ξW ] and 

∇ W− r P , ∇hW + 2∇ W− r e

2

J0 h e

2

ξ

are differential operators acting on weighted spaces that preserve the weights. This implies first that   

P ∇ξW X = ∇ξW (P X) + P , ∇ξW X

 = ∇ξW (κ2 ) + ∇ξW (T2 ) + P , ∇ξW X. From the special structure of κ2 , the first term in the right-hand side of the last equation belongs to C2∞ , the middle ∞ , and the last one to C ∞ . From the estimations given above on the critical weights of P and [4, §I.2], term to C1+δ  2 ∞ for some δ  > 0. Arguing similarly, this ensures that ∇ξW X ∈ C1+δ   

 P ∇hW X = ∇hW (P X) + P , ∇hW X  −r W  e 2 ∇ξ X + QX, = ∇hW (κ2 ) + ∇hW (T2 ) − 2∇ W− r e

2

J0 h

. where Q is a weight-preserving operator. This is precisely what we need to conclude that ∇hW X ∈ C ∞ 3  2 +δ

We can now rewrite (2.7) in a slightly more precise manner: LX g0 = LX (g0 − g) ¯ + LX g¯ ¯ − Q(X, g) ¯ + κ e−r + h = LX (g0 − g) = κ e−r +S3 ,

(2.9) κ e−r ;

Cρ∞3

as a result, S3 then lives in for some ρ3 > 2. Choosing a where we have collected in S3 all terms except r r − − −r 2 2 local g0 -orthonormal basis (∂r , e ξ, e v, e J0 v) in a collar neighbourhood of type [R0 , +∞[ × U , we can also write the vector field X as r

r

X = a∂r + e−r bξ + c e− 2 v + d e− 2 J0 v where a, b, c, d are functions belonging to weighted spaces with the same weight as X does. Using the explicit form of the Levi-Civita connection of g0 exhibited in [5, Lemma 2.1], we can expand the expression of LX g0 . Keeping only its dominant part as the left-hand side, Eq. (2.9) yields ⎞ ⎛ 3b a d b c a 2 − 2 + 2 2 −c 2 −d  c ⎟ ⎜ b − 3b + a 0 − d2 ⎟ ⎜2 2 2 2 (2.10) ⎟ = κ e−r + S4 ⎜ c d a − c − − 0 ⎠ ⎝ 2 2 2 d 2

−d

c 2

0

− a2 r

where a prime denotes differentiation with respect to r and S4 lives in Cρ∞4 for some ρ4 > 2 (as ∇ g¯ = ∇ W + O(e− 2 ), tangential derivatives for the Levi-Civita connection of g0 are of decay faster than 2, so that they can be collected in the S4 -term). Looking carefully at the symmetric matrix on the left-hand side in (2.10), one notices that its restriction to H has no anti-J0 -linear part that could equal the κ-term in the right-hand side which is the only term of decay e−2r . Then, κ necessarily vanishes identically and the decay of the matrix coefficients is controlled by the S4 -term. This argument then ends the proof. 2 Remark 2.5. Note that another important consequence of Proposition 2.4 is the uniqueness of the formally undetermined term k e−r in (2.4) once a contact form η has been fixed. We shall now define the renormalized volume:

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Proposition 2.6. Let g be an ACHE metric on M. Then the volume of large “balls” M(r) of radius r (complements of ]r, +∞[ × X in M) in the metric ψ ∗ g has an asymptotic expansion:  1 2r θ ∧ dθ + v1 er +V + o(1), vol M(r) = e 2 X

where v1 is a formally determined term. To check the proposition, just notice that the volume form of g only differs from that of g¯ at order 52 since k is trace-free, and, in the volume form of the Kähler form ω, order 32 terms do not exist whereas order 2 terms are of zero integral from the CR Stokes’ formula [8,15]: whenever α = α1 θ 1 is a (1, 0)-form on X (given in a local orthonormal coframe), one has ¯

¯

dα = α1,1¯ θ 1 ∧ θ 1 + α1,0 η ∧ θ 1 + α1 τ1¯1 η ∧ θ 1 (recall α·,· denotes the components of ∇α in the local coframe), and    ¯ α1,1¯ η ∧ θ 1 ∧ θ 1 = (dα) ∧ η = − α ∧ dη = 0, X

X ¯ = iθ 1 ∧ θ 1 .

since X is closed and dη has the form  dr ∧ η ∧ dη, with

X

For future reference, we note that the order e−2r term in the volume expansion

  1 ¯ (2.11) R − 4i τ1¯ 1 ,11 − τ1 1 ,1¯ 1¯ . 12 The number V is the renormalized volume of the metric g associated to the choice of pseudo-hermitian structure at infinity, and also (up to now) to the Kähler-adapted coordinates given by ψ . To check directly that it is independent of the choice of coordinates (i.e., in the language above, of the choice of ψ ), consider two choices of diffeomorphisms ψ1 , ψ2 as above such that =

(ψi )∗ g = g¯ + ki e−r + o(e−2r ), for i = 1, 2 and ki are in the relevant sub-bundle of symmetric bilinear forms. Then, in a collar neighbourhood of infinity, Proposition 2.4 applies to ψ2 ◦ψ1 −1 : it is of the form “identity plus a weighted term that decays to 0 at infinity” in any system of uniform charts around infinity. As (M, g0 ) has controlled sectional curvature and volume growth around infinity, its injectivity radius has a positive lower bound, hence one concludes that ψ2 ◦ ψ1 −1 (p) = expp X(p) for some uniquely defined vector field X, with weighted control.  Proposition 2.4 implies that X decays as fast as e−(2+δ )r for some δ  > 0, and this is enough to ensure that the function r ◦ ψ −1 is uniquely defined up to terms of fast enough decay by the requirement that the metric g¯ in Corollary 2.3 be Kähler–Einstein at order 2 at least, and that the ACHE metric g be asymptotic to it in the way expressed in (2.4). This achieves the proof of Theorem 1.1. Remark 2.7. In the AHE case, the renormalized volume is similarly defined [12], but with a different choice of coordinates: given a metric h∞ in the conformal class of the boundary at infinity, one can always find a function ρ such that g = dρ 2 + h(ρ) on ]ρ0 , +∞[ × X and lim∞ e−2ρ h(ρ) = h∞ (the function ρ is the geodesic defining function associated to the choice of h∞ in the conformal class at infinity). The metric h(ρ) has an expansion in powers of e−ρ  can be defined as above as the constant coefficient in the expansion of vol(M(ρ)). and a renormalized volume V As it is in this case an invariant of g in dimension 4, the reader might hence think that the “misbehaviour” of the renormalized volume in the ACHE case comes from a bad choice of coordinates. However, the metric of CH2 in geodesic coordinates is given by:   ρ γ0 , gCH2 = dρ 2 + 4 sinh2 (ρ)η0 2 + 4 sinh2 2

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and easy computations lead to a volume development of geodesic balls:   π 2 2ρ e −2π 2 eρ +3π 2 + o(1) vol B(ρ) = 2  = 3π 2 . Moreover, χ = 1 and the bulk curvature integrals vanish: hence, up to the mandatory factor 3 2 , so that V 8π  alone cannot complement the renormalized Gauss–Bonnet formula and some extra contribution is needed to get a V correct formula. Choosing now the r-coordinate to be adapted to the complex structure J as in Corollary 2.3 leads to a different presentation:     4 4 gCH2 = 1 − e−2r (dr 2 + e2r η0 ) + 1 − 2 e−r + e−2r er γ0 + o(e−2r ) 3 3 5

2 obtained from the previous formula by letting ρ = r + 13 e−2r + O(e− 2 r ). One computes V = 10 3 π , and this again cannot be the only extra term in the Gauss–Bonnet formula. At last, note moreover that Seshadri’s definition of the renormalized volume [18] also presents the same features.

Remark 2.8. The other important fact to be noted in Proposition 2.6 is that there is no term in vol M(r) that is linear in r, i.e.  L =  η ∧ dη = 0. X

In the AHE case, when the boundary at infinity is odd-dimensional, an analogous phenomenon occurs: the linear term vanishes pointwise in the asymptotic expansion of the volume form [12]. In the AHE case again, but when the boundary at infinity is even-dimensional, a linear term appears in the expansion of the volume form, with a coefficient related to the Q-curvature of the boundary at infinity [12]. If one believes in the analogy between even-dimensional conformal geometry and CR geometry, one might then wonder why there is indeed no linear term in Proposition 2.6. Reasoning by analogy again, one would assert from [12] that the coefficient of the linear term should be a multiple of the integral on X of the CR Q-curvature defined by Fefferman and Hirachi [11], but this always vanishes in dimension 3. This is indeed what happens as the linear term of (2.11) differs from QCR by a (non-zero) divergence term, as expected. The CR case of dimension 3 stands then in an intermediate position between the odd- and even-dimensional conformal cases: the volume form expansion does exhibit a linear term, but it integrates to zero, so that the volume expansion has none. The linear term also appears in Seshadri’s work [18] (it is called there a ‘log term’, as he uses a reference radial function s ∼ e−r ) and he shows that its vanishing after integration is a purely 3-dimensional phenomenon. Seshadri’s linear term is proportional to the one given by (2.11), as one could expect: both his and our radial functions are presumably related by a non-singular development (no log/linear terms), so that the first singular terms in the volume forms’ expansions necessarily have to agree. 3. Proof of Theorem 1.2 We first choose a contact form (or pseudo-hermitian structure) at infinity (i.e. on X) realizing the CR structure. Choosing a diffeomorphism ψ such that (2.4) holds, and using what has been proved above, we assume from now on that the ACHE metric g itself satisfies (2.4) (rather than ψ ∗ g), so that no diffeomorphism is involved. The basic element of the proof then is the Gauss–Bonnet–Chern formula for the Euler characteristic of the compact domain with boundary M(r) delimited by what we shall call the coordinate sphere S(r) = {r} × X:       1 1 1 2 2 Scal2 Scal + |W | − χ M(r) = 24 8π 2 96π 2 M(r)

1 + 12π 2



M(r)

T (I ∧ I ∧ I) + 3T (I ∧ R), S(r)

(3.1)

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where W and Scal denote the Weyl and scalar curvatures of (M, g) (trace-free Ricci curvature is zero as g is Einstein), the ∧ operation provides a p + q-form with values in ⊗r+s T M from a p-form with values in ⊗r T M and a q-form with values in ⊗s T M, and we have denoted by T the contraction between the volume form of S(r) and elements of ⊗3 T M. Moreover, I is the shape operator of S(r) in (M, g) and R is the curvature (2-form with values in 2-forms) of (M, g). Notice also the difference in notation between the Tanaka–Webster curvature R of the 3-dimensional pseudohermitian manifold X and the curvature tensor R of the 4-dimensional Einstein manifold M. 1 It is proven in [5] that the integral involving |W |2 − 24 Scal2 on M(r) converges for an ACHE metric when r goes to infinity. Moreover, it is clear that both the scalar curvature integral (which is, up to a constant, vol M(r)) and the boundary integrals have an asymptotic expansion in powers of e−r/2 (there are no polynomial terms as they cancel in the volume form expansion, as noted above). Convergence of all the other terms implies that divergent terms cancel pairwise, whereas the limit as r goes to infinity of    1 1 3 2 2 χ(M) − − V Scal |W | − 24 8π 2 8π 2 M(r)

is given by the constant terms in the asymptotic expansion of the boundary integrals. Our task then reduces to a careful computation of these terms. For this, the following facts will be useful: Fact 1. It is proven in [5] that replacing g by g¯ in the boundary integrals introduces terms that are o(e−2r ) only, hence do not contribute in the limit as the volume form of each sphere grows like e2r . Hence all computations can be done using the formal Kähler–Einstein metric g¯ rather than the actual ACHE metric g. This has the important consequence that the boundary contribution is necessarily an integral of a tensorial expression in the coefficients of g, ¯ i.e. an integral of a local pseudo-hermitian invariant of the boundary at infinity. Fact 2. As said above, we only need to track the zeroth-order terms in the boundary integral. Using d volg0 = e2r η ∧dη as a reference volume form, this amounts to look for the e−2r -terms in the integrands. In the course of the computations, every function we will meet has an asymptotic expansion of the form: 3

A0 + A1 e−r + A 3 e− 2 r + A2 e−2r + o(e−2r ), 2

where all terms are measured with respect to g0 . Terms in e−2r (“order 2 terms”) may then only occur when putting together an order 2 term with order 0 terms or two order 1 terms with order 0 terms. Order 32 terms can hence be forgotten, unless some differentiation is to be performed later, as derivating along H raises the order by a factor 12 . Fact 3. Changing η to λ2 η shows that the boundary contribution is the integral of a local scalar pseudohermitian invariant of weight 4 (i.e. is multiplied by λ−4 in this process). It is well-known that any such invariant in dimension 3 is a linear combination of R 2 and |τ |2 only [19]. Hence one should find a result of the type:   3 2 χ(M) = bulk integral + V + C R η ∧ dη + C |τ |2 η ∧ dη, 1 2 8π 2 X

X

and the universal constants C1 and C2 are the only data that have to be determined below. Looking back at the 1 1 1¯ coefficients of g¯ in Corollary 2.3, one sees that every term in R,11¯ , R,11 ¯ , R = R,11¯ + R,11 ¯ , τ,0 , τ1,11 ¯ , τ1,1¯ 1¯ or ¯

1 1 R,0 = τ1,11 + τ1, ¯ 1¯ 1¯ (Bianchi identity) indeed drops out, using the CR Stokes’ formula already described in the proof of Proposition 2.6. In what follows, occurrence of any such “irrelevant” term will be denoted by the notation O.

It then remains to compute constants C1 and C2 . In principle, both could be computed by considering enough examples, but this involves identifying some ACHE metrics and computing the bulk integrals in the Weyl curvature. With the notable exception of the complex hyperbolic plane itself, this task is not obviously easier than handling directly the boundary contribution in (3.1). We shall then compute directly the boundary term in (3.1), keeping only in mind terms in R 2 or |τ |2 . The example of CH2 treated above can be used at the end as a cross-check for the value of C1 , but not for C2 .

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The task is divided into three steps: computation of the outer unit normal and intrinsic volume form of S(r), computation of the shape operator (the only step that involves differentiation, but, as we shall see, this is totally harmless) and estimation of the order 2 terms in T (I ∧ I ∧ I) and T (I ∧ R). The computations are rather long but otherwise straightforward, and we give only the intermediate results. Step 1. From the explicit expansion of g, ¯ we immediately get the outer unit normal of S(r):    2 1 R ν(r) = 1 + e−2r (3.2) − |τ |2 + O ∂r + ν T + o(e−2r ) 3 8 where ν T is an order

3 2

¯

1 1 term tangent to X, involving linearly R,1 , R,1¯ , τ1,1 ¯ and τ1,1¯ . The volume form  of S(r) is

then (up to forgotten order

3 2

terms, see Fact 3 above):

1  = ιν(r) ω2 2 = e2r (0 + e−r 1 + e−2r 2 ) + o(e−2r )    2 1 R R η ∧ dη + o(e−2r ). − |τ |2 + O = e2r 1 − e−r + e−2r 2 3 8

(3.3)

¯ Step 2. The shape operator I is obtained by taking the extrinsic covariant derivative of the unit outer normal ∇ν(r) (where ∇¯ here denotes the Levi-Civita connection of g). ¯ As ν T is an order 32 term, only its derivatives in the direction 1 1¯ , hence and τ1, of H might contribute to order 2 terms in I, but these would add only terms linear in R,11¯ , R,11 ¯ , τ1,11 ¯ 1¯ 1¯

are O-terms in the sense of Fact 3. We shall then neglect the occurrence of the term ν T , and this simplifies a lot the computations. It remains to compute the derivative of the radial term in the unit normal, which we still denote by ν(r). If we had a precise description of the Levi-Civita connection of g, ¯ this would be a simple task, but we haven’t (see [5] for the first term, which is not enough for our needs). We then first seek an expression for the bilinear symmetric form g( ¯ ∇¯ · ν(r), ·)|T S(r) . Since we shall end with a symmetric bilinear form, we may keep only symmetric terms in the usual 6-term formula for the Levi-Civita covariant derivative:       2g¯ ∇¯ U ν(r), V = ν(r) · g(U, V ) + U · g ν(r), V − V · g ν(r), U   

  

  − g [U, V ], ν(r) + g U, ν(r) , V − g V , ν(r) , U . The only symmetric term is the first one, and it is given explicitly by:    1 −2r R 2 1 2 ¯ 1+ e − |τ | + O ∂r g. 2 3 8 This is easily evaluated from the expansion of g¯ recalled above and one gets   1 R ¯ g¯ ∇ν(r), · = e2r η2 + er γ + e−r γ (J0 τ · ,·) 2 2     1 R2 1 R2 2 2 2 + − |τ | η − − |τ | e−r γ + O. 3 8 6 8

(3.4)

Using the explicit expression of g¯ once more and computing one step further immediately yields the (endomorphism) shape operator: 3

I = I0 + e−r I1 + e− 2 r I 3 + e−2r I2 + o(e−2r ), 2

(3.5)

where 1 R I1 = IdH −J0 τ, I0 = Idξ + IdH , 2 4  2  2   R R 5 I2 = − |τ |2 Idξ + + |τ |2 IdH +O 8 16 2 and the precise value of I 3 is irrelevant as before. 2

(3.6)

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Step 3. We now obtain the desired contributions of the integral terms in formula (3.1) by chasing the order 2-terms. The contraction T (I ∧ I ∧ I) is explicitly described as follows: if symmetric endomorphisms A, B, and C are diagonal in a basis {e1 , e2 , e3 } of S(r) chosen to be g0 -orthonormal, with eigenvalues αu , βs and γt , and if  = f 0 then: T (A ∧ B ∧ C) = f S(αu βs γt )0 ,

(3.7)

where S denotes the sum over all permutations of {u, s, t}. r r We now perform the computations in a basis (∂r , e−r ξ, e− 2 h, e− 2 J0 h), with h a γ -unit element of H chosen to be an eigenvector of the symmetric endomorphism J0 τ . This basis is orthogonal for g0 and we recall that the relevant terms in I are given by (3.6), where we denote: I2 = A Idξ +B IdH +O, with R2 R2 5 2 B= − |τ |2 , + |τ | . 8 16 2 The term T (I ∧ I ∧ I) is now the sum of contributions of type a , Ib ∧ Ic ∧ Id , with a + b + c + d = 2; the results are easily obtained and are given by the following table: A=

Involved terms

Result

2 , I0 ∧ I0 ∧ I0 sum of the 3 terms in 1 , I1 ∧ I0 ∧ I0 sum of the 3 terms in 0 , I1 ∧ I1 ∧ I0 sum of the 3 terms in 0 , I2 ∧ I0 ∧ I0

1/16R 2 − 1/2|τ |2 −3/4R 2 3/2R 2 − 6|τ |2 3/2A + 6B = 9/16R 2 + 27/2|τ |2

For instance, the first term is recovered as follows: the order 2-term in  is   1 R2 2 = − |τ |2 η ∧ dη 3 8 whereas the order 0-term I0 has eigenvalues α = 1 (of multiplicity 1, in the ξ -direction) and β = in the plane H ), so that the resulting contribution in T (I ∧ I ∧ I) is

1 2

(of multiplicity 2,

2 , I0 ∧ I0 ∧ I0  = S(αββ)2 = 6αβ 2 2   1 1 1 2 2 =6× × R − |τ | η ∧ dη 4 3 8   1 2 1 2 R − |τ | η ∧ dη. = 16 2 The other computations are entirely similar. For the evaluation of the T (I ∧ R)-term, the contraction T acts as follows: if A is a diagonal endomorphism, with eigenvalues λt , in the same basis as above and if ρ is an endomorphism on 2-forms with constant coefficients and with diagonal entries Krs = ρ(er ∧ es ), er ∧ es , then: T (A ∧ ρ) = A(Krs λt ) ,

(3.8)

where A denotes the sum over circular (not all) permutations of {r, s, t}. We now need to describe the curvature tensor R of g¯ (seen as a 2-form with values in 2-forms) a little bit further from [5]. It is given as a sum of three terms: the model curvature tensor (i.e. that has the same expression w.r.t. (g, ¯ J ) as the constant holomorphic sectional 2 curvature has w.r.t. g CH , J0 ), an order 2 term called W2− that is a multiple of the Cartan tensor of the CR-structure at infinity, and a higher order term. In short, R = R0 + W2− e−2r + o(e−2r ).

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From Fact 3 above, terms formed from the Cartan tensor cannot have the correct homogeneity, hence cannot appear in the final formula. The complicated W2− -term may then be forgotten. This makes now easy the evaluation of all necessary terms, although the computations are slightly more complicated than the previous ones, due to the presence of the algebraically more involved curvature term; the results are given in the following table: Involved terms

Result

2 , I0 ∧ R0 1 , I1 ∧ R0 0 , I2 ∧ R0

−5/96R 2 + 5/12|τ |2 1/16R 2 −A − 1/2B = −5/32R 2 − 1/4|τ |2

Once again, we give as an example the details of the first computation and leave the other ones to the reader: the model curvature tensor R0 has eigenvalues K23 = −1 on the 2-form defining the H -plane and K1i = − 12 (i = 2, 3) on orthogonal 2-forms containing η. Using the form of the order 2-term 2 already described, this leads to the following first contribution to T (I ∧ R): 2 , I0 ∧ R0  = (K12 λ3 + K13 λ2 + K23 λ1 )2     1 1 1 1 2 = − − −1 × R − |τ |2 η ∧ dη 8 8 3 8   5 2 5 2 = − R + |τ | η ∧ dη. 96 12 One can now conclude the entire set of computations by collecting together all intermediate results listed in the two tables. The first contribution is     5 7 1 6 15 3 5 − R2 + − |τ |2 = − R 2 + |τ |2 , T (I ∧ R) = − + 96 96 96 12 12 48 6 and the second is



T (I ∧ I ∧ I) =

   1 1 1 12 6 9 27 − + + R2 + − − 6 + |τ |2 = − R 2 + 7|τ |2 . 16 16 16 16 2 2 4

According to formula (3.1), Theorem 1.2 is obtained by adding the second result to three times the first. 4. Examples and final comments The complex hyperbolic case revisited. Using the computations done in Remark 2.7, we can check the constant 2 in front of the R 2 -term. Indeed, one has χ = 1, V = 10 3 π and, using the conventions of [5],  η0 ∧ dη0 = π 2 . R = 4 and

1 16

S3

An easy computation then shows that this is consistent with our result. The case of domains. If our ACHE manifold is a connected strictly pseudo-convex domain in C2 , the metric g can be taken to be Kähler–Einstein (hence equals g). ¯ This metric is known as the Cheng–Yau metric of the domain [9]. It is interesting to compare in this case our formula (1.1) with the renormalized characteristic class defined in [5] (see also [6]). Namely, for any ACHE manifold (M, g) with boundary at infinity X,  1 1 3|W − |2 − |W + |2 + Scal2 = χ(M) − 3τ (M) + ν(X), (4.1) 2 24 8π M

where ν(X) is a CR invariant of X. For a domain Ω with the Cheng–Yau metric, this reduces to  3 |W − |2 = χ(Ω) + ν(X) 8π 2 M

(4.2)

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(note that the signature of a domain is zero since the compactly supported cohomology of C2 always vanishes). Comparing (1.1) with (4.2) yields 3 (4.3) V = 2χ(Ω) − ν(X), 4π 2 where V is the invariant defined in (1.2). In this special case, we then note the interesting feature that the renormalized volume is, up to a integral topological contribution, an invariant purely of the boundary at infinity. It has moreover in this case a locally determined first derivative with respect to variations of Ω (whereas it is easy to check that its derivative at a non-Kähler–Einstein ACHE metric g with respect to general variations of J0 does depend on the formally undetermined term k of Eq. (2.4)). Comparison with Seshadri’s results. In [18], N. Seshadri has defined another version of the renormalized volume for asymptotically complex hyperbolic Kähler–Einstein metrics in every dimension. For instance, this covers the Cheng–Yau metrics on domains in Cm . He then derives a renormalized Gauss–Bonnet formula, which, in complex dimension m = 2, reads as follows:     1 2 6 c2 − c1 = 2 VS + S(η) η ∧ dη χ(M) − (4.4) 3 π M

X

where VS stands for Seshadri’s version of the renormalized volume, and S(η) is a function on X, depending on the choice of a pseudo-hermitian structure η, and which is a linear combination of R 2 and |τ |2 . Formulas (1.1) and (4.4) are quite close, but are not entirely identical. The integral term in M in (1.1) reduces to the integral term in (4.4) in the Kähler–Einstein case, but the others do not match exactly. This seemingly surprising discrepancy can be explained as follows: a renormalized Gauss–Bonnet formula is obtained whenever one compares the (divergent) integrand of the polynomial in the curvature corresponding to some topological invariant with a convergent one, if the difference between the two involves only a constant times the volume form (see [1] for a lucid description in the AHE case). Hence, the renormalized volume appears in both cases after integration of a squared scalar curvature term (this is (c1 )2 in Seshadri’s approach). However, we choose the Einstein equation Ricg¯ = − 32 g¯ (scalar curvature equal to −6) and the volume form of g¯ to be 12 ω2 , whereas N. Seshadri relies on complex analysis techniques for which it is usual to normalize the Ricci form ρ¯ of g¯ to ρ¯ = −3ω (which makes scalar curvature equal to −12) and he takes the volume form to be 12 ( 2i )2 ω2 . These different conventions then explain why renormalized volumes appear with different coefficients in front in formulas (1.1) and (4.4). Moreover, a second difference comes from our different choices of radial functions for the expansion of the volume forms, that lead to slightly different definitions of V . It is however very likely that each defining function can be computed from the other by a power series that is formally determined at all relevant terms, so that the results differ by local terms on the boundary at infinity only. Acknowledgements The author thanks Olivier Biquard, Gilles Carron and Jean-Marc Schlenker for their interest in this work, and C. Robin Graham for useful comments. He is also very grateful to the referee for his in-depth analysis of the paper and his useful criticisms and suggestions. References [1] [2] [3] [4] [5] [6] [7]

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