SCATTERING AND INVERSE SCATTERING ON ACH MANIFOLDS 1. Introduction Scattering ...

38 Pages·2008·0.44 MB·English

SCATTERING AND INVERSE SCATTERING ON ACH MANIFOLDS
COLINGUILLARMOUANDANTOˆNIOSA´ BARRETO
Abstract. We study scattering and inverse scattering theories for asymptotically complex
hyperbolic manifolds. We show the existence of the scattering operator as a meromorphic
familyof operators inthe Heisenbergcalculus on the boundary, which isa contact manifold
with a pseudohermitian structure. Then we define the radiation fields as inthe real asymp-
totically hyperbolic case, and reconstruct the scattering operator from those fields. As an
applicationweshowthatthemanifold,includingitstopologyandthemetric,aredetermined
uptoinvariantsbythescatteringmatrixatallenergies.
1. Introduction
Scatteringtheoryandinverseproblemsforrealasymptoticallyhyperbolicmanifoldhavebeen
extensively studied, see for example [15, 19, 20, 27, 34, 35, 40] and references cited there. The
purpose of this work is to extend to asymptotically complex hyperbolic manifolds, ACH in
short,severalresultsinscatteringandinversescatteringwhichareknownforrealasymptotically
hyperbolic manifolds,
The class of ACH manifolds studied here was introduced by Epstein, Melrose and Mendoza
[8], and it contains certain quotients of the complex hyperbolic space by discrete groups, as
well as smooth pseudo-convex domains in Cn+1 equipped with a K¨ahler metric of Bergman
type. More recently similar classes of manifolds have also been considered by Biquard [5] and
Biquard-Herzlich [6].
Before discussing asymptotically complex hyperbolic manifolds, we recall certain facts and
results about real asymptotically hyperbolic manifolds, which we believe are helpful to under-
stand the complex case. An (n+1)-dimensional non-compact manifold X equipped with a C∞
Riemannian metric g is called asymptotically hyperbolic if it compactifies into a C∞ manifold
X¯ with boundary ∂X¯, and if ρ is a defining function of the boundary ∂X¯, ρ2g is a C∞ met-
ric which is non-degenerate up to ∂X¯, and moreover if dρ = 1 at ∂X¯. It can be shown,
ρ2g
| |
see [14], that (X,g) is asymptotically hyperbolic if and only if there exists a diffeomorphism
ψ :[0,ǫ) ∂X¯ U X¯ with ψ( 0 ∂X¯)=∂X¯ such that
t
× → ⊂ { }×
dt2+h(t)
(1.1) ψ∗g =
t2
whereh(t),t [0,ǫ)isasmooth1-parameterfamilyofC∞ metricson∂X¯.Thefunctionρ:=ψ t
∗
isaboundary∈definingfunctioninX¯ near∂X¯,whichcanbeextendedsmoothlytoX¯.Notethat
the boundary represents the geometric infinity of X, as does the sphere Sn for the hyperbolic
space Hn+1.
The spectrum of ∆ , the Laplacian of (X,g) was studied in [34]; it consists of a finite pure
g
point spectrum σ (∆), which is the set of L2(X) eigenvalues, and an absolutely continuous
pp
spectrum σ (∆) satisfying
ac
σ (∆)= n2/4, and σ (∆) 0,n2/4 .
ac pp
∞ ⊂
(cid:2) (cid:1) (cid:0) (cid:1)
2000 Mathematics Subject Classification. Primary58J50, Secondary35P25.
Key words and phrases. scattering,pseudoconvex domains,asymptoticallycomplexhyperbolicmanifolds.
Thefirstauthor waspartiallysupportedbyFrenchANRgrantsno. [JC05-52556] and[JC05-46063], andan
Australian National University postdoctoral fellowship. Both authors were supported in part by the National
ScienceFoundationGrantDMS0500788.
1
2 COLINGUILLARMOUANDANTOˆNIOSA´ BARRETO
The resolvent
R(λ)=(∆ λ(n λ))−1,
g
− −
which is a bounded operator in L2(X), for (λ) > n, has a finite meromorphic extension to
C (n 1)/2 N , where N = Nsup 0 , asℜa map fr2om C∞(X) C∞(X), see [34, 18]. The
\ − − 0 0 { } 0 →
polesofR(λ)arecalledresonances. Here,andthroughoutthepaper,wecallafamilyofoperators
finite-meromorphic ifitis meromorphic,i.e. ithasafinite Laurentexpansionateachpoint,and
the rank of the polar part at a pole has finite rank. The finite meromorphic continuation of
R(λ)totheentirecomplexplaneexistsifandonlyifh(t)hasanevenTaylorexpansionatt=0.
If h(t) is not even, R(λ) might have essentialsingularities at the points (n 1)/2 N , see [18].
0
It has been shown in [15, 27] that for (λ) = n/2, (λ) = 0, and any−f C−∞(∂X¯), there
ℜ ℑ 6 ∈
exists a unique u C∞(X) satisfying
λ
∈
(∆ λ(n λ))u =0
g λ
− −
such that near ∂X¯
uλ =ρn−λf−+ρλf++O(ρn2+1), f− =f, f+ C∞(∂X¯).
∈
One can use this to define the scattering operator S(λ), for (λ) = n/2, (λ) = 0, as a
ℜ ℑ 6
generalized Dirichlet-to-Neumann map
S(λ):C∞(∂X¯) C∞(∂X¯)
−→
f f .
+
7−→
Like the Dirichlet-to-Neumann map, S(λ) is an elliptic pseudo-differential operator, but of
order 2λ n. It extends meromorphically to C. The first author recently studied Krein theory
−
in even dimension manifolds by introducing a generalizeddeterminant of S(λ) and applied it to
analyze the Selberg zeta function for certain quotient of hyperbolic space by discrete groups of
isometries, in continuation of work by Patterson-Perry [37]. The second author studied inverse
problemsusing S(λ). He firstprovedwith Joshi[27]that S(λ) for λ fixeddetermines the Taylor
expansionofh(t)in(1.1),thenmorerecentlyheprovedin[40]thatthemapλ S(λ)determines
→
the whole manifold up to global isometry.
We give a precise definition of what we call an asymptotically complex hyperbolic metric in
Section 3, but we will briefly explain this notion before stating our results. We consider a
non-compact Riemannian manifold (X,g) that compactifies into X¯ smooth with boundary ∂X¯.
We assume that the boundary admits a contact form Θ and an almost complex structure
0
J :kerΘ kerΘ suchthatdΘ (.,J.) is symmetric positive definite onkerΘ .The associated
0 0 0 0
→
Reeb vector field T is the one which satisfies
0
Θ (T )=1 and dΘ (T ,JZ)=0 for any Z kerΘ .
0 0 0 0 0
∈
The parabolic dilation M on T∂X¯ =kerΘ RT is defined by
ρ 0 0
⊕
M (V +tT )=ρV +ρ2tT , V kerΘ ,t R.
ρ 0 0 0
∈ ∈
Wesaythat(X,g)isACHifthereexistsadiffeomorphismφ:[0,ǫ) ∂X¯ φ([0,ǫ) ∂X¯) X¯
ρ ρ
such that φ( 0 ∂X¯)=∂X¯ and × → × ⊂
{ }×
4dρ2+dΘ (.,J.) Θ2 4dρ2+h(ρ)
φ∗g = 0 + 0 +ρQ =:
ρ2 ρ4 ρ ρ2
for some symmetric tensors Q on ∂X¯ satisfying M∗Q C∞([0,ǫ) ∂X¯,S2(T∗∂X¯)). We
ρ ρ ρ ∈ ρ ×
call such a φ a product decomposition and we say that g is even at order 2k if h−1(ρ) has
only even powers in its Taylor expansion at ρ = 0 at order 2k, here h−1(ρ) is the metric
dual to h(ρ) on T∗∂X¯. We will show that the latter is independent of φ. Note that if ρ is
any boundary defining function in X¯, then ρ4g|∂X¯ = e4ω0Θ20 for some ω0 ∈ C∞(∂X¯), thus
we have more naturally a conformal class of 1-form [Θ ] on the boundary induced by g. We
0
call the boundary ∂X¯ equipped with (Θ ,J) a pseudo-hermitian structure and its conformal
0
class ([Θ ],J) a conformal pseudo-hermitian structure. On such a manifold, one can define the
0
SCATTERING AND INVERSE SCATTERING ON ACH MANIFOLDS 3
classψ∗ (∂X¯)ofHeisenbergpseudo-differentialoperatorsassociatedtothe contactdistribution
Θ0
kerΘ and its related principal symbol, see [9, 39, 7] and Subsection 4.3 below. We can define
0
the “parabolically homogeneous norm” on T∂X¯:
1 1
V := Θ (V)2+ dΘ (V,JV)2 4.
He 0 0
|| || (cid:16) 4 (cid:17)
and the metric h :=M∗(ρ−2h(ρ)) =dΘ (.,J.)+Θ2.
0 ρ |ρ=0 0 0
As in the real asymptotically hyperbolic manifolds, the spectrum of the Laplacian ∆ of an
g
ACH manifold (X,g) consists of an absolutely continuous part and the pure point spectrum
satisfying
(1.2) σ (∆ )= (n+1)2/4, and σ (∆ ) 0,(n+1)2/4 ,
ac g pp g
∞ ⊂
(cid:2) (cid:1) (cid:0) (cid:1)
where σ (∆ ) is a finite set of eigenvalues. The resolvent
pp g
n+1
R(λ):=(∆ λ(n+1 λ))−1 L(L2(X)), (λ)> ,
g
− − ∈ ℜ 2
is meromorphic,and Epstein, Melrose and Mendoza [8] provedthat it has a finite-meromorphic
extensiontoC (P N ),whereP := n+1/2 1N ,asafamilyofpseudo-differentialoperators
in a certain ca\lculu0∪s.−If w0e assume0that g i2s ev−en2 a0t order 2k, k N , it will be shown that P
0 0
∈
may be replaced by P := n+1/2−k 1N .
k 2 − 2 0
As in the real case, we use this strong result to show that for any λ with (λ) = (n+1)/2,
and (λ)=0, and for any f C∞(∂X¯), there exists a unique u C∞(X) sℜatisfying
λ
ℑ 6 ∈ ∈
(∆ λ(n+1 λ))u =0,
g λ
− −
such that near ∂X¯
uλ =ρn+1−λf−+ρλf++O(ρn+21+1), f− =f, f+ C∞(∂X¯).
∈
We then define the scattering operator
S(λ):C∞(∂X¯) C∞(∂X¯)
−→
f f .
+
7−→
This operator depends on the first derivative of ρ at ∂X¯, and thus equivalently on the con-
formal representative in [Θ ]. For another choice ρˆ= eωρ of boundary defining function with
0
ω C∞(X¯), we clearly have the conformal covariance
∈
Sˆ(λ)=e−2ω0S(λ)e2(n+1−λ)ω0, ω0 :=ω|∂X¯.
The structure of the operator S(λ) is established in
Theorem 1.1. Let (X,g) be an ACH manifold which is even at order 2k, k N, then the
scattering operator S(λ) extends to C ( N P ) as a meromorphic family o∈f conformally
0 k
\ − ∪
covariant operators in the class of the Heisenberg pseudodifferential operators Ψ4λ−2(n+1)(∂X¯),
Θ0
which is unitary on L2(∂X¯,dvol ) when (λ) = n+1 and (λ) = 0. The principal symbol of
h0 ℜ 2 ℑ 6
S(λ) is
22λ+1Γ(λ)2
σ (S(λ))(ξ)=c F ( V −4λ)
pr nΓ(2λ n 1) V→ξ || ||He
− −
wherec CdependsonlyonnandF denotesFouriertransformfromT∂X¯ toT∗∂X¯.Moreover,
n
S(λ) is fi∈nite-meromorphic in C ( N P (n+1 P )) and has at most poles of order 1
0 k k
at each λ := n+1 + 1k with k \N,−the r∪esidu∪e of whic−h is a Heisenberg differential operator in
k 2 4 ∈
Ψk (∂X¯)plusafiniterankprojectorappearing ifandonlyifλ (n 1 λ ) σ (∆ ). Moreover
Θ0 k − − k ∈ pp g
at λ , we have
2k
k
1
Res S(λ)= ( ∆ +i(k+1 2l)T ) mod Ψ2k−1(∂X¯)
λ2k 2((k 1)!k!) − b − 0 Θ0
− Yl=1
4 COLINGUILLARMOUANDANTOˆNIOSA´ BARRETO
where ∆ ,T are the horizontal sublaplacian and the Reeb vector field of (∂X¯,Θ ,J).
b 0 0
We also deduce from [19] an explicit formula between finite-multiplicity poles of S(λ) (scat-
tering poles) and finite multiplicity poles of R(λ) (resonances)in Proposition6.5 and show that
essential singularities for S(λ) and R(λ) can occur at (n+1/2 N )/2 if the metric has no
0
−
evenness property, see Proposition 6.7.
TheproofthatS(λ), (λ)= n+1, (λ)=0,isapseudodifferentialoperatorintheHeisenberg
ℜ 2 ℑ 6
calculus is sketched by Melrose in [36]. The novelties in this theorem are the computation of
the principal symbol of S(λ), its meromorphic continuation, and the analysis of the poles. In
the case where the manifold X¯ is a strictly pseudoconvex domain of Cn+1 equipped with an
approximate Einstein K¨ahler metric, the relationship between the residues Res S(λ) and the
λ2k
Gover-Grahamoperators of [12] is announced in [24].
WealsostudythescatteringtheoryfromadynamicalviewpointasintheLax-Phillipstheory.
Wedefinetheradiationfields,showthattheygiveunitarytranslationrepresentationsofthewave
group which can be used to define the scattering matrix (6.1) from the wave equation.
The Cauchy problem for the wave equation
(n+1)2
D2 ∆ u(t,z)=0 in R X
(1.3) (cid:18) t − g − 4 (cid:19) +×
u(0,z)=f (z), D u(0,z)=f (z), f ,f C∞(X)
1 t 2 1 2 ∈ 0
has smooth solutions u C∞(R X), we consider the behavior of u at infinity along some
+
∈ ×
bicharacteristics and prove
Theorem1.2. Letz =(ρ,z′) [0,ǫ) ∂X¯ besomecoordinatesgivenbyaproductdecomposition
φ as above. Let u(t,z) be the s∈olution×of (1.3) near ∂X¯, then
v (ρ,s,z′):=ρ−n−1u(s 2logρ,ρ,z′) C∞([0,ǫ) R ∂X¯).
+
− ∈ × ×
We define the forward radiation field as the operator
R :C∞(X) C∞(X) C∞(R ∂X¯),
+ 0 × 0 −→ ×
(1.4) ∂
(f ,f ) v (0,s,z′).
1 2 +
7−→ ∂s
Similarly one can show that v (ρ,s,z′):=ρ−n−1u(s+logρ,ρ,z′) is smooth on [0,ǫ) R ∂X¯
−
and we can define the backward radiation field by R (f ,f ):=∂ v (0,s,z′). × ×
− 1 2 s −
Let H1(X)= f L2(X); df L2(X) and let E :=Π (H1(X) L2(X)) where Π is
g ac ac ac
{ ∈ | | ∈ } ×
the orthogonal projection from L2(X) onto the space of absolute continuity of ∆ . The space
g
E is a Hilbert space when equipped with the norm . defined by
ac E
||||
1 (n+1)2
(ω ,ω ) 2 := (dω 2 ω 2+ ω 2) dvol .
|| 0 1 ||E 2Z | 0| − 4 | 0| | 1| g
X
Then we show
Theorem 1.3. The forward and backward radiation fields R extendtoisometric isomorphisms
±
from E to L2(R ∂X¯,drdvol ). Moreover, the map defined by
ac × h0
(1.5) S:=R R−1 :L2(R ∂X¯,drdvol ) L2(R ∂X¯,drdvol )
+ − × h0 → × h0
is unitary and is a convolution operator in s, and conjugating it with Fourier transform in s we
have
FSF−1(λ)= S(λ).
−
The operator S in (1.5) is the dynamical definition of the scattering operator. Next, using
these tools, and after proving a localization result for the support of functions f L2 (X) for
which R (0,f)=0 in s ( ,s ) (a “support theorem” in the sense of Helgason∈[22a,c23] and
+ 0
∈ −∞
Lax-Phillips [29]), we are able to prove the following result on inverse scattering:
SCATTERING AND INVERSE SCATTERING ON ACH MANIFOLDS 5
Theorem 1.4. Let (X ,g ),(X ,g ) be two ACH manifolds with the same boundary M :=
1 1 2 2
∂X¯ = ∂X¯ , and equipped with the same conformal class of contact forms [Θ ] = [Θ ]. Let
1 2 0,1 0,2
S (λ) and S (λ) be the corresponding scattering operators associated to a conformal representa-
1 2
tive Θ [Θ ]. If S (λ)=S (λ) on (λ)= n+1, λ=0 , then there exists a diffeomorphism
0 ∈ 0,1 1 2 {ℜ 2 ℑ 6 }
Φ:X¯ X¯ such that Φ=Id on M and Φ∗g =g .
1 2 2 1
→
The method we use is very close to that introduced by the second author [40] in the asymp-
totically hyperbolic case, which was inspired by the boundary control theory of Belishev [3].
The paper is organized as follows: In Section 1, we consider the model case of the complex
hyperbolic space Hn+1, then we discuss the geometry of ACH manifolds near infinity in Section
C
2. We review the Θ-calculus of Epstein-Melrose-Mendoza [8] on X, and define the Heisenberg
calculuson∂X¯ (thesearethe“natural”classesofpseudo-differentialoperatorsassociatedtothe
geometric structure) in Section 4, and we analyze the Poisson and the scattering operators in
Section 6. The next sections consist in defining radiation fields (Section 7), prove their relation
withscatteringoperator(Section8),the supportTheorem(Section10)andthe inverseproblem
(Section 11). We conclude with a technical appendix.
Acknowledgement We thank D. Geller, R. Graham, P. Greiner, M. Olbrich and R. Ponge
for helpful discussions.
2. The model case of Hn+1
C
2.1. Hn+1 and the Heisenberg group H . Thehyperbolic complexspaceofcomplexdimen-
C n
sion n+1 is denoted by Hn+1, it is the unit ball Bn+1 = z Cn+1; z < 1 equipped with
C
the K¨ahler metric g := 4∂∂¯log(ρ) where ρ := 1 z 2. N{ot∈e that ρ|is|a bo}undary defining
0
− −| |
function of the closed complex ball. The holomorphic curvature is 1 and this metric is called
the Bergman metric. Another model of Hn+1 is given by z Cn+1−;Q(z,z)>0 where Q is
C
{ ∈ }
i 1
(2.1) Q(z,z′)= (z z¯′) z z¯′,
−2 1− 1 − 2 j j
Xj>1
and the boundary (the sphere S2n+1) is a compactification of the Heisenberg group
1
H := z Cn+1;Q(z,z)=0 = ( (z ), ω 2,ω);(z ,ω) Cn+1 R Cn R2n+1,
n 1 1
{ ∈ } { ℜ 2| | ∈ }≃ × ≃
thus Hn+1 (0, ) H . The variable u:= (z ) is the one lying in R and we have a contact
C n 1
≃ ∞ × ℜ
form on H given by
n
Φ:=du+y.dx x.dy,
−
where ω =x+iy Cn =Rn+iRn. The functions
∈
ρ0 :=Q(z,z)21, u= (z1), ω Cn
ℜ ∈
give coordinates on (0, ) H Hn+1 and the Bergman metric with holomorphic curvature
n C
∞ × ≃
1 is given in this model by
−
4dρ2+2dω 2 Φ2
(2.2) g = 0 | | + .
0 ρ2 ρ4
0 0
The Heisenberg group H is a Lie group with the law
n
(u,ω). (u′,ω′):=(u+u′ (ω.ω¯′),ω+ω′),
Hn −ℑ
theoriginis0andtheinverse(u,ω)−1 =( u, ω). AbasisfortheLiealgebrah ofH isgiven
n n
− −
by the left invariant vector fields
1 1
(2.3) X = (∂ y ∂ ), Y = (∂ +x ∂ ), T =∂ .
j √2 xj − j u j √2 yj j u u
6 COLINGUILLARMOUANDANTOˆNIOSA´ BARRETO
The map (u,ω) R Cn uT + (ω )X + (ω )Y identifies H with h , and the group
∈ × 7→ jℜ j j ℑ j j n n
law becomes P
W. W′ =(Φ(W +W′) dΦ(W,W′))T +π (W +W′),
Hn − kerΦ
where π is the projection on kerΦ parallel to T.
kerΦ
ThecomplexhyperbolicspacehasaLiegroupstructure,thisisactuallyasemi-directproduct
of the multiplicative group ((0, ), ) with (H ,. ). We introduce the parabolic dilation
M (ρ ,u,ω) := (δρ ,δ2u,δω) on∞(0,× ) H (hnereHnδ > 0), then the group law on Hn+1
δ 0 0 n C
∞ × ≃
(0, ) H is
n
∞ ×
(2.4) (ρ ,W). (ρ′,W′):=(ρ ρ′,W. M (W′)).
0 HnC+1 0 0 0 Hn ρ0
andwehaveforthislaw(ρ ,W)−1 =(ρ−1, M W). Itiseasytocheckthatthecorresponding
0 0 − ρ−01
Lie algebra has a basis
(2.5) ρ ∂ ,ρ2∂ ,ρ X ,...,ρ X ,ρ Y ,...,ρ Y ,
0 ρ0 0 u 0 1 0 n 0 1 0 n
which is orthonormal with respect to the metric g . This algebra will be denoted by ΦT Hn+1,
0 0 C
to agree with the notation used in the next sections. Observe also that these vectors and the
metric g are homogeneous of degree 0 under the parabolic dilation M .
0 δ
2.2. The Resolvent kernel for Hn+1. The spectrum of the Bergman Laplacian ∆ of Hn+1
C g0 C
is absolutely continuous and equal to σ(∆ ) = [(n+1)2, ), this leads to study the modified
g0 4 ∞
resolvent
R(s):=(∆ s(n+1 s))−1
g0 − −
which is bounded on L2(Hn+1,dvol ), provided (s) > n+1. The Schwartz kernel of R(s)
C g0 ℜ 2
has been computed by Epstein-Melrose-Mendoza[8] and admits a meromorphiccontinuationto
C, with poles at N of finite multiplicity (contrary to what is written in [8]). By symmetry
0
−
arguments,thiskernelR(s;z;z′)isexpressedasafunctionoftheBergmandistanceofd (z;z′).
g0
We have
d (z;z′) Q(z,z′)
cosh g0 = | |
(cid:18) 2 (cid:19) (Q(z,z)Q(z′,z′))12
where Q is defined in (2.1). Using a polar decomposition around the diagonal, the kernel
R(s;z,z′) is obtained as a solution of an hypergeometric ODE, exactly like in the real case,
and is given by
Γ(s)2
R(s;z,z′)=c r(z;z′)s F (s,s,2s n;r(z;z′)), where
nΓ(2s n) 2 1 −
(2.6) −
d (z;z′) −2 4ρ2ρ′2
r(z;z′):= cosh g0 = 0 0 ,
(cid:18) (cid:18) 2 (cid:19)(cid:19) (u u′+ (ω.ω¯′))2+(ρ2+ρ′2+ 1 ω ω′ 2)2
− ℑ 0 0 2| − |
with c constant depending on n and F is a hypergeometric function (see [2]), we also used
n 2 1
the formula
i 1 1
Q(z,z′)= − (u u′+ (ω.ω¯′))+ ρ2(z)+ρ′2(z)+ ω ω′ 2 .
2 − ℑ 2(cid:16) 0 0 2| − | (cid:17)
A change of variables shows that if an operator K has a distributional Schwartz kernel which
is of the form k(r(z,z′)), in other words, it depends only on d (z,z′), then K is a convolution
g0
operator with respect to the group law on Hn+1:
C
4µ2 2n+1dµdtdz
(2.7) Kf(ρ ,u,ω)= k f (ρ ,u,ω). (µ,t,z)−1 ,
0 Z (cid:18)t2+(1+µ2+ 12|z|2)2(cid:19) (cid:16) 0 HnC+1 (cid:17) µ
where µ−1dµdtdz is a right invariant mesure. The resolvent kernel (2.6) is of this form (s is a
parameter), so the action of the operator R(s) on a function is given by (2.7).
SCATTERING AND INVERSE SCATTERING ON ACH MANIFOLDS 7
Remark: We see that the poles at m N have residue
0
− ∈−
m m
P = a rk−m =(Q(z,z)Q(z′,z′))−m a Q(z,z′)2m−2k(Q(z,z)Q(z′,z′))k
m m,k m,k
| |
kX=0 Xk=0
forsomea C. ButclearlyP hasfiniteranksinceitisapolynomialtimesQ(z,z)−mQ(z′,z′)−m.
m,k m
∈
So the poles are of finite multiplicity.
3. Asymptotically complex hyperbolic manifolds
3.1. Θ metrics. WestartbydescribingtheΘstructuresofEpstein-Melrose-Mendoza[8],which
generalizeBergmantypemetricsonpseudoconvexdomains,aswellasquotientsΓ Hn+1ofHn+1
C C
\
by convex co-compact groups of isometries.
Let X¯ =X ∂X be a smooth 2n+2-dimensional compact manifold with boundary ∂X¯ and
letΘ C∞(∂X¯∪,T∗X¯)be asmooth1-formon∂X¯ suchthatifi:∂X¯ X¯ isthe inclusion,then
Θ :=∈i∗Θ does not vanish on ∂X¯. According to the terminology of [→39], the boundary ∂X¯ has
0
the structure of a Heisenberg manifold equipped with the subbundle kerΘ .
0
We first recall a few definitions introduced in [8]. If ρ is a boundary defining function of ∂X¯,
we define the Lie subalgebra V of C∞(X¯,TX¯) by the condition
Θ
V V V ρC∞(X¯,TX¯),Θ(V) ρ2C∞(X¯),
Θ
∈ ⇐⇒ ∈ ∈
where Θ C∞(X¯,T∗X¯) is any smooth extension of Θ.eIt is shown in [8] that V only depends
Θ
on the co∈nformal class of Θ. Let T,N,Y ,...,Y be a smooth local frame in X¯ near a point
p ∂X¯esuch that 1 2n
∈
Span(N,Y ,...,Y ) kerΘ, Span(T,Y ,...,Y ) T∂X¯, dρ(N)=Θ(T)=1.
1 2n 1 2n
⊂ ⊂
Then any V V can be writtenenear p as e
Θ
∈
2n
(3.1) V =aρN +bρ2T + c ρY , a,b,c C∞(X¯)
i i i
∈
Xi=1
and
(3.2) ρN,ρ2T,ρY ,...,ρY
1 2n
formabasisofV overC∞(X¯)nearp. TheLiealgebraV isthesetofsmoothsectionsofavector
Θ Θ
bundle over TX¯, we denote by ΘTX¯ this bundle. Let F be the set of vector fields vanishing
p
at p if p X or the set of vector fields of the form (3.1) satisfying a(p) = b(p) = c (p) = 0 if
i
p ∂X¯. ∈The fibre ΘT X¯ at p X¯ can be defined by ΘT X¯ := V /F . If p ∂X¯, ΘT X¯ is a
p p Θ p p
Li∈e algebra, and any vector v ∈ΘT X¯ can be represented as ∈
p
∈
2n
v =aρN +bρ2T + c ρY , a,b,c R.
i i i
∈
Xi=1
The dual bundle ΘT∗X¯ of ΘTX¯ has for local basis near p ∂X¯ the dual basis to (3.2)
∈
dρ Θ α α
1 2
, , ,..., .
ρ ρ2 ρ ρ
e
A Θ-metric is a smooth positive symmetric 2-tensor on ΘT∗X¯
g C∞(X¯,S2(ΘT∗X¯)).
∈ +
We are interested in the special cases of Θ-metrics for which Epstein, Melrose and Mendoza [8]
proved the meromorphic extension of the resolvent. We begin by the first assumption, which
allows one to find particular boundary defining functions.
8 COLINGUILLARMOUANDANTOˆNIOSA´ BARRETO
3.2. Modelboundarydefiningfunctions. LetgbeaΘ-metric,itthusrestrictstoastandard
metric in the interior X. If ρ is a boundary defining function, we can define the vector field
X V as the dual of dρ/ρ via the metric g, i.e. g(X ,v) = ρ−1dρ(v) for any v ΘTX¯, this
ρ Θ ρ
is a∈smooth non-vanishing section of ΘTX¯. It is clear that in X, we have ∈
gρ X
ρ2gρ= ∇ = ρ
∇ ρ2 ρ
which extends to a non-vanishing vector in C∞(X¯,TX¯) transverse to ∂X¯ since dρ(X /ρ) =
ρ
dρ/ρ2 =0 on ∂X¯ (here means gradient). We first assume that
| |g 6 ∇
1
(H1) X = on ∂X¯
ρ g
| | 2
and it is easy to check that this condition does not depend on ρ. The restriction ρ4g|T∂X¯ is
conformal to the tensor Θ2, which leads to the definition of the conformal class [Θ ].
0 0
bLoeumndmaray3d.e1fi.nLinetgef2uωn0cΘti0on∈[ρΘ0o]fw∂iXt¯h ωsu0c∈hCth∞a(t∂XX¯), th=en1th/e2reinexaistnseaighubnoiuqurhe,ooudpotof C∂X0∞¯(Xan)d,
ρ g
| |
ρ4g|T∂X¯ =e4ω0Θ20.
Proof: If x is a boundary defining function we search for a function ω C∞(X¯) such that
ρ:=eωx satisfies X 2 = dρ/ρ2 =1/4 near ∂X¯. This can be rewritten un∈der the form
| ρ|g | |g
(3.3) 2Xx(ω) + |dω|2g = 1/4−|Xx|2g.
x x x
Since dω 2 = O(x2), X = 1/2 at x = 0, this is a first order non-linear PDE with smooth
| |g | x|g
coefficients, and it is easy to check that it is non-characteristic since X /x is transverse to ∂X¯.
x
By prescribing the value ω =ω , we obtain a unique solution in a neighbourhoodof ∂X¯. (cid:3)
x=0 0
|
Such a boundary defining function will be called a model boundary defining function. Let φ
t
be the flow of the vector field 4X /ρ, we consider the diffeomorphism
ρ
[0,ǫ) ∂X¯ φ([0,ǫ) ∂X¯) X¯
φ: × → × ⊂
(cid:26) (t,y) φt(y)
→
Then ρ(φ (y))=t and for any Z T∂X¯
t
∈
4 dt(Z)
φ∗g(∂ ,∂ )= , φ∗g(∂ ,Z)= =0.
t t t2 t t2
We will write t = ρ and X = ρ∂ for what follows and we call this diffeomorphism a prod-
ρ ρ
uct decomposition near ∂X¯. Note also that Θ(∂ ) = 0 since ρ∂ V . With this product
ρ ρ Θ
∈
decomposition, the metric g has the form
4dρ2+h(ρ)
(3.4) g =
ρ2
in (0,ǫ) ∂X¯ with h(ρ) a family of metrics on T∂X¯ for ρ = 0 and such that h(ρ)/ρ2
ρ
C∞(∂X¯,S×2(ΘT∗X¯)) depending smoothly on ρ [0,ǫ). 6 ∈
We will say that the metric is even if h(ρ)∈−1, as a metric on T∗∂X¯ has an even Taylor
expansionatρ=0intheproductdecomposition. Itisstraightforwardtoseethatthiscondition
is invariant with respect to the choice of model boundary defining function ρ (i.e. of product
decomposition), for instance from the proof of Lemma 2.1 in [18] where the PDE is replaced in
our case by (3.3). Indeed, if x is a model boundary defining function and ρ=eωx another one,
ω has to satisfy (3.3), that is
2∂ ω+x (∂ ω)2+ dω 2 =0
x (cid:16) x | |h(x)(cid:17)
SCATTERING AND INVERSE SCATTERING ON ACH MANIFOLDS 9
and the evenness of the Taylor expansion of dω 2 at x = 0 was all that we needed in [18].
| |h(x)
Note that evenness at order 2k can also be defined invariantly by requiring∂2j+1h−1(0)=0 for
ρ
all j <k (see again [18] for similar definition in the real case).
3.3. Additional assumptions. Following [8], we define for p ∂X¯ the one-dimensional sub-
space of ΘT X¯ ∈
p
K := V ρ2C∞(X¯,TX¯) /F
2,p p
{ ∈ }
and the 2n dimensional subspace
K := V V ;V =ρW,W tangent to ∂X¯ /F .
1,p Θ p
{ ∈ }
The subspace K is a two-step nilpotent Lie algebra which is the fibre over p of the tangent
1,p
Lie bundle defined in [39]. We denote by K ,K the bundles over ∂X¯ whose fibre at p are
1 2
K ,K . Near p ∂X¯, let (Y ,...,Y ) be a local basis of kerΘ T∂X¯ and T T∂X¯ such
1,p 2,p 1 2n 0
that Θ (T) = 1, th∈is give a local basis of T∂X¯. A basis of K is⊂given by the ∈class of ρ2T
0 2,p
mod F , whereas (ρY ,...,ρY ,ρ2T) mod F gives a basis of K . This easily shows that K
p 1 2n p 1,p 2
is included in the centre of K .
1
Let us denote K = ker(ρ−1dρ) the subbundle of ΘTX¯, it is isomorphic to T∂X¯ over ρ = 0
1
and equal to K over ρ=0. Thus the choice of a function ρ (or product decomposition of ∂6 X¯)
1 e
induces orthogonal decompositions for g (outside ρ=0 for the first one)
{ }
TX¯ R∂ T∂X¯, ΘTX¯ Rρ∂ K .
ρ ρ 1
≃ ⊕ ≃ ⊕
Using this decomposition, we extend Θ0 on R∂ρ T∂X¯ to be coenstant with respect to ρ, and
⊕
suchthatΘ (∂ )=0,in particularΘ is extended by φ∗Θ at ρ=0 . Thenρ−2Θ is asmooth
0 ρ 0 0
{ }
section of ΘT∗X¯ and kerΘ ,ker(ρ−2Θ ) are respective subbundle of T∂X¯,K . We have an
0 0 1
isomorphism of vector bundles
e
(T∂X¯/kerΘ ) kerΘ K
ψ : 0 ⊕ 0 → 1 ,
(cid:26) (p;Tp Yp) (p;ρY +ρ2T mod Fp)
⊕ →
whereY,T aresmoothlocalsectionsofT([0,ǫ) ∂X¯),constantwithrespecttoρ,suchthatY
ρ
kerΘ , Y(p)=Y , and T(p)=T mod kerΘ . ×Via ψ , the form Θ on (T∂X¯/kerΘ ) kerΘ∈
0 p p 0 ∗ 0 0 0
is mapped onto the form ρ−2Θ on K . The subbundle (T∂X¯/kerΘ ) is mapped ⊕onto K
0 1 0 2
by ψ and kerΘ onto the bundle ker(ρ−2Θ ). Similarly the 2-form dΘ is mapped onto
(ρ−2dΘ ) 0 . Alocalchoiceofvector0T transversaltokerΘ inT0∂|Xk¯eriΘn0aneighbourhood
0 |ker(ρ−2Θ0) 0
Up of p∈∂X¯ fixes a vector ρ2T transversalto ker(ρ−2Θ0), thus a representative vector ρ2T|∂X¯
ofK ,andalocalbasisρ−1α ,...,ρ−1α fortheannihilator(ker(ρ−2Θ ))∗ ofRρ2T inK∗ (i.e.
2 1 2n 0 1
the dual of ker(ρ−2Θ )) can be chosen.
0 e
Inview ofthis discussion,we havethatρ−2h(ρ) C∞([0,ǫ) ∂X¯,S2(K∗)) andwecanwrite
near a point p ∂X¯ ∈ × + 1
∈ e
h(ρ) Θ2 2n α α 2n α Θ
(3.5) =a 0 + c i⊗ j + b i⊗ 0
ρ2 ρ4 ij ρ2 i ρ3
iX,j=1 Xi=1
for some functions a,b ,c C∞(X¯). Note also that a is globally defined and can be taken
i ij ρ=0
∈ |
to be 1 by changing the conformal representative of [Θ ].
0
Let us denote by g the metric on ΘT X¯, in terms of (3.4) and (3.5), this is
p p
dρ2 Θ2 2n α α 2n α Θ
(3.6) g =4 + 0 + c (p) i⊗ j + b (p) i⊗ 0.
p ρ2 ρ4 ij ρ2 i ρ3
iX,j=1 Xi=1
The assumptions of [8] on the metric correspond to the minimal assumptions for which g is
p
isometric to the complex hyperbolic metric. The second assumption made in [8] is that
(H2) Θ is a contact form on ∂X¯
0
10 COLINGUILLARMOUANDANTOˆNIOSA´ BARRETO
which means that dΘ is non-degenerate on kerΘ . The next hypothesis is that for the orthog-
0 0
onal decomposition K =K L for g , the map
1,p 2,p p p
⊕
L ΘT X¯
µ: p → p
(cid:26) Z [ρ∂ρ,Z]
→
is the identity. Since L is spanned by some Z = l ρY +k ρ2T mod F the assumption
p i j ij j i p
clearly reduces to k = 0 since ρ2T mod F commPutes with any elements of K , therefore
i p 1,p
L =(ker(Θ /ρ2)) . Thenbyorthogonalityofthe decomposition,this meansthatb (p)=0,i.e.
p 0 p i
Θ
0
(H3) ker K .
(cid:16)ρ2(cid:17)⊥g 2
The last assumption of [8] is
Θ dΘ Θ
(H4) J End ker 0 ,J2 = Id and 0(.,J.)=g on ker 0 K
∃ ∈ (cid:16) (cid:16)ρ2(cid:17)(cid:17) − ρ2 (cid:16)ρ2(cid:17)⊂ 1
which, using the bundle isomorphism ψ, is actually equivalent to the following
(H4’) J End(kerΘ ), J2 = Id and dΘ (.,J.)=ρ2g
∃ ∈ 0 − 0 |kerΘ0
where the restriction ρ2g is the metric on the bundle kerΘ T∂X¯ whose value on fiber
|kerΘ0 0 ⊂
(kerΘ ) is the limit lim t2(φ∗g) .
0 p t→0 (t,p)
3.4. Asymptotically complex hyperbolic manifolds. An aymptotically complex hyperbolic
manifold, or ACH manifold, is a non-compact Riemannian manifold (X,g) such that there
exists a smooth compact manifold with boundary X¯ which compactifies X, equipped with a
Θ-structure, such that g is a Θ-metric satisfying assumptions (H1) to (H4).
In view of the above discussion there exists a product decomposition (0,ǫ) ∂X¯ near the
ρ
×
boundary where the metric can be expressed by
4dρ2+h(ρ)
(3.7) g =
ρ2
with h(ρ) a smooth family of metrics on ∂X¯ for ρ=0 such that
6
h(ρ) Θ2 dΘ (.,J.)
(3.8) = 0 + 0 +ρg, g C∞(X¯,S2(ΘT∗X¯)).
ρ ρ4 ρ2 ∈
The form Θ induces the metric h and the veolumee density on ∂X¯
0 0
(3.9) h :=Θ2+dΘ (.,J.), dvol = Θ dΘn .
0 0 0 h0 | 0∧ 0|
By choosing a different representative Θˆ0 = eω0Θ0, it is easy to check that the corresponding
nmaetturricalistohˆ0ca=llet2hωe0Θpa20i+r (e[Θω0d]Θ,J0)(.a,Jc.o)nafnodrmthael pvsoeluudmoeheformrmitiiasndvsotrlhˆu0ct=uree(no+n1∂)ωX0¯d.volh0. It is then
0
In view of the assumptions on Θ for an ACH manifold, there exists a smooth global vector
0
field, denoted by T , tangent to ∂X¯ such that Θ (T ) = 1 and dΘ (T ,Y) = 0 for every
0 0 0 0 0
Y kerΘ ; this is Reeb’s vector field. With the notation of (3.7), we can define
0
∈
M∗h(ρ)
ρ
(3.10) k(ρ):= ,
ρ2
where M : T∂X¯ T∂X¯ is the dilation M (tT +V) := δ2tT +δV if V kerΘ , t R and
δ δ 0 0 0
T is Reeb’s vecto→r field. Observe that k(ρ) is a smooth family of metrics ∈on ∂X¯ up t∈o ρ = 0,
0
k(0)=h and the volume form is dvol =ρ−2n−2dρdvol =ρ−2n−3dρdvol .
0 g h(ρ) k(ρ)
Remark: AnACHmanifoldinthesenseofBiquard-Herzlich[6]isquitesimilartooursetting,
thedifferenceliesinthetermgof(3.8): forthem,g =O(ρδ)forsomeδ >0andgdoesnothavea
polyhomogeneousexpansionattheboundary,whereasinourcasethemetricispolyhomogeneous
but we can allow terms of ordeer O(ρ−3) in the Θedirection, for instance. e
0