Sharp estimates of the Kobayashi metric and Gromov hyperbolicity

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by  Florian Bertrand

SHARP ESTIMATES OF THE KOBAYASHI METRIC AND GROMOV HYPERBOLICITY
FLORIANBERTRAND
8 ABSTRACT. LetD = {ρ < 0}beasmoothrelativelycompact domaininafour dimensional almost com-
0 plex manifold (M,J), where ρ is a J-plurisubharmonic function on a neighborhood of D and strictly J-
0 plurisubharmoniconaneighborhoodof∂D. WegivesharpestimatesoftheKobayashimetric. Ourapproach
2 isbasedonanasymptoticquantitativedescriptionofboththedomainDandthealmostcomplexstructureJ
near a boundary point. Following Z.M.Balogh and M.Bonk [1], these sharp estimates provide the Gromov
n
hyperbolicityofthedomainD.
a
J

3

]
V
C
INTRODUCTION
.
h
t Onecandefinedifferentnotionsofhyperbolicityonagivenmanifold,basedongeometricstructures, and
a
it seems natural to try to connect them. For instance, the links between the symplectic hyperbolicity and
m
the Kobayashi hyperbolicity were studied by A.-L.Biolley [3]. In the article [1], Z.M.Balogh and M.Bonk
[
established deep connections between the Kobayashi hyperbolicity and the Gromov hyperbolocity, based

1
on sharp asymptotic estimates of the Kobayashi metric. Since the Gromov hyperbolicity may be defined
v
on any geodesic space, it is natural to understand its links with the Kobayashi hyperbolicity in the most
5
0 generalmanifoldsonwhichtheKobayashimetriccanbedefined,namelythealmostcomplexmanifolds. As
5 emphasizedby[1],itisnecessarytostudypreciselytheKobayashimetric. Sincethereisnoexactexpression
0
of this pseudometric, except for particular domains where geodesics can be determined explicitely, we are
.
1 interestedintheboundary behaviouroftheKobayashimetricandinitsasymptoticgeodesics. Onecannote
0
thatboundaryestimatesofthisinvariantpseudometric,whoseexistenceisdirectlyissuedfromtheexistence
8
0 of pseudoholomorphic discs proved by A.Nijenhuis-W.Woolf [21], isalso afundamental tool for the study
: oftheextension ofdiffeomorphisms andfortheclassification ofmanifolds.
v
i ThefirstresultsinthisdirectionareduetoI.Graham[12],whogaveboundaryestimatesoftheKobayashi
X
metricnearastrictlypseudoconvexboundarypoint,providingthe(local)completehyperbolicitynearsucha
r
a point. ConsideringaL2-theoryapproach,D.Catlin[5]obtainedsimilarestimatesonpseudoconvexdomains
of finite type in C2. A crucial progress in the strictly pseudoconvex case is due to D.Ma [19], who gave
an optimal asymptotic description of this metric. His approach is based on a localization principle given
by F.Forstneric and J.-P.Rosay [9] using some purely complex analysis arguments as peak holomorphic
functions. Theestimates proved byD.Mawereused in[1]to provethe Gromovhyperbolicity ofrelatively
compactstrictlypseudoconvex domains. TheaimofthispaperistoobtainsharpestimatesoftheKobayashi
metriconstrictlypseudoconvex domainsinfouralmostcomplexmanifolds:
Theorem A. Let D be a relatively compact strictly J-pseudoconvex smooth domain in a four dimensional
almost complex manifold (M,J). Then for every ε > 0, there exists 0 < ε < ε and positive constants C
0
2000MathematicsSubjectClassification. 32Q45,32Q60,32Q65,32T25,32T15,54E35.
Keywordsandphrases. Almostcomplexstructure,Kobayashimetric,Gromovhyperbolicspace.
1
2 FLORIANBERTRAND
andssuchthatforeveryp D N (∂D)andeveryv = v +v T M wehave
∈ ∩ ε0 n t ∈ p
1
e−Cδ(p)s 4|δv(np|)22 + LJρ2(πδ((pp)),vt) 2 ≤ K(D,J)(p,v)
(cid:18) (cid:19)
1
(0.1) eCδ(p)s |vn|2 + LJρ(π(p),vt) 2 .
≤ 4δ(p)2 2δ(p)
(cid:18) (cid:19)
In the above theorem, δ(p) := dist(p,∂D), where dist is taken with respect to a Riemannian metric.
Forpsufficiently close tothe boundary the point π(p)denotes theunique boundary point such that δ(p) =
p π(p) . MoreoverN (∂D) := q M,δ(q) < ε . Wepointoutthatthesplittingv = v +v T M
k − k ε0 { ∈ 0} n t ∈ p
intangent andnormalcomponents in(0.1)isunderstood tobetakenatπ(p).
Asacorollary ofTheoremA,weobtain:
TheoremB.
(1) LetD bearelatively compactstrictly J-pseudoconvex smoothdomain inanalmost complex mani-
fold(M,J)ofdimensionfour. ThenthedomainDendowedwiththeKobayashiintegrateddistance
d isaGromovhyperbolic metricspace.
(D,J)
(2) Each point in a four dimensional almost complex manifold admits a basis of Gromov hyperbolic
neighborhoods.
Thepaper is organized as follows. In Section 1, wegive general facts about almost complex manifolds.
InSection2,weshowhowtodeduceTheoremBfromTheoremA.Finally,Section3isdevotedtotheproof
ofourmainresult,namelyTheoremA.
1. PRELIMINARIES
Wedenoteby∆theunitdiscofCandby∆ thediscofCcentered attheoriginofradiusr > 0.
r
1.1. Almostcomplexmanifoldsandpseudoholomorphicdiscs. AnalmostcomplexstructureJ onareal
smooth manifold M is a (1,1) tensor field which satisfies J2 = Id. We suppose that J is smooth. The
pair(M,J)iscalledanalmostcomplexmanifold. WedenotebyJ−thestandardintegrablestructureonCn
st
forevery n. Adifferentiable mapf : (M ,J ) (M,J) between twoalmost complex manifolds is said
′ ′
tobe(J ,J)-holomorphic ifJ(f(p)) d f = d−→f J (p),foreveryp M . IncaseM = ∆ C,such
′ p p ′ ′ ′
◦ ◦ ∈ ⊂
amapiscalledapseudoholomorphic disc. Iff : (M,J) M isadiffeomorphism, wedefineanalmost
′
−→
complexstructure, f J,onM asthedirectimageofJ byf:
′
∗
f J(q) := df−1(q)f J f−1(q) dqf−1,
∗ ◦ ◦
foreveryq M . (cid:0) (cid:1)
′
∈
Thefollowing lemma (see [10]) states that locally any almost complex manifold can be seen as the unit
ballofCn endowedwithasmallsmoothpertubation ofthestandard integrable structure J .
st
Lemma 1.1. Let (M,J) be an almost complex manifold, with J of class k, k 0. Then for every
C ≥
point p M and every λ > 0 there exist a neighborhood U of p and a coordinate diffeomorphism
0
z : U ∈ B centered a p (ie z(p) = 0) such that the direct image of J satisfies z J(0) = J and
st
→ ∗
z (J) J λ .
|| ∗ − st||Ck(B¯) ≤ 0
SHARPESTIMATESOFTHEKOBAYASHIMETRICANDGROMOVHYPERBOLICITY 3
Thisissimplydonebyconsideringalocalchartz : U Bcenteredap(iez(p) = 0),composingitwith
→
alineardiffeomorphism toinsurez J(0) = J anddilating coordinates.
st
SoletJ beanalmostcomplex st∗ructure definedinaneighborhood U oftheorigin inR2n,andsuch that
J issufficientlyclosedtothestandardstructureinuniformnormontheclosureU ofU. TheJ-holomorphy
equation forapseudoholomorphic discu: ∆ U R2n isgivenby
→ ⊆
∂u ∂u
(1.1) J(u) = 0.
∂y − ∂x
According to [21], for every p M, there is a neighborhood V of zero in T M, such that for every
p
∈
v V,thereisaJ-holomorphic discusatisfying u(0) = pandd u(∂/∂x) = v.
0
∈
1.2. Splitting of the tangent space. Assume that J is a diagonal almost complex structure defined in a
neighborhood of theorigin inR4 and such that J(0) = J . Consider abasis (ω ,ω )of(1,0) differential
st 1 2
formsforthestructure J inaneighborhood oftheorigin. SinceJ isdiagonal, wemaychoose
ω =dzj B (z)dz¯j, j = 1,2.
j j
−
Denoteby(Y ,Y )thecorresponding dualbasisof(1,0)vectorfields. Then
1 2
∂ ∂
Y = β (z) , j = 1,2.
j ∂zj − j ∂zj
Moreover B (0) = β (0) = 0 for j = 1,2. The basis (Y (0),Y (0)) simply coincides with the canonical
j j 1 2
(1,0) basis of C2. In particular Y (0) is a basis vector of the complex tangent space TJ(∂D) and Y (0)
1 0 2
is normal to ∂D. Consider now for t 0 the translation ∂D t of the boundary of D near the origin.
≥ −
Consider, in a neighborhood of the origin, a (1,0) vector field X (for J) such that X (0) = Y (0) and
1 1 1
X (z) generates the J-invariant tangent space TJ(∂D t) at every point z ∂D t, 0 t << 1.
1 z − ∈ − ≤
Setting X = Y , we obtain a basis of vector fields (X ,X ) on D (restricting D if necessary). Any
2 2 1 2
(1,0)
complex tangent vector v T (D,J) at point z D admits the unique decomposition v = v + v
z t n
∈ ∈
where v = α X (z) is the tangent component and v = α X (z) is the normal component. Identifying
t 1 1 n 2 2
(1,0)
T (D,J) with T D we may consider the decomposition v = v +v for each v T (D). Finally we
z z t n z
∈
consider thisdecomposition forpointsz inaneighborhood oftheboundary.
1.3. Levigeometry. Letρbea 2 realvaluedfunctiononasmoothalmostcomplexmanifold(M,J).We
C
denotebydcρthedifferential formdefinedby
J
(1.2) dcρ(v) := dρ(Jv),
J −
wherev isasection ofTM. TheLeviformofρatapointp M andavectorv T M isdefinedby
p
∈ ∈
ρ(p,v) := d(dcρ)(p)(v,J(p)v) = ddcρ(p)(v,J(p)v).
LJ J J
In case (M,J) = (Cn,J ), then ρis, up to apositive multiplicative constant, the usual standard Levi
st LJst
form:
∂2ρ
ρ(p,v) = 4 v v .
LJst ∂z ∂z j k
j k
X
WeinvestigatenowhowcloseistheLeviformwithrespecttoJ fromthestandardLeviform. Forp M
∈
andv T M,weeasilyget:
p
∈
(1.3) ρ(p,v) = ρ(p,v)+d(dc dc )ρ(p)(v,J(p)v)+ddc ρ(p)(v,J(p) J )v).
LJ LJst J − Jst Jst − st
4 FLORIANBERTRAND
Inlocalcoordinates (t ,t , ,t )ofR2n,(1.3)maybewrittenasfollows
1 2 2n
···
ρ(p,v) = ρ(p,v)+tv(A tA)J(p)v+t(J(p) J )vDJ v+
LJ LJst − − st st
(1.4) t(J(p) J )vD(J(p) J )v
st st
− −
where
∂u ∂J ∂2u
i,j
A:= and D := .
∂t ∂t ∂t ∂t
Xi i k !1 j,k 2n (cid:18) j k(cid:19)1≤j,k≤2n
≤ ≤
Letf bea(J ,J)-biholomorphism from(M ,J )to(M,J). Thenforeveryp M andeveryv T M:
′ ′ ′ p
∈ ∈
J′ρ(p,v) = Jρ f−1(f(p),dpf (v)).
L L ◦
Thisexpresses theinvariance oftheLeviformunderpseudobiholomorphisms.
Thenextproposition isusefulinordertocomputetheLeviform(see[16]).
Proposition 1.2. Letp M andv T M. Then
p
∈ ∈
ρ(p,v) =∆(ρ u)(0),
J
L ◦
whereu:∆ (M,J)isanyJ-holomorphic discsatisfying u(0) = pandd u(∂/∂ ) = v.
0 x
→
Proposition 1.2leadstothefollowingproposition-definition:
Proposition 1.3. Thetwostatementsareequivalent:
(1) ρ uissubharmonic foranyJ-holomorphic discu: ∆ M.
◦ →
(2) ρ(p,v) 0foreveryp M andeveryv T M.
J p
L ≥ ∈ ∈
IfoneofthepreviousstatementsissatisfiedwesaythatρisJ-plurisubharmonic. Wesaythatρisstrictly
J-plurisubharmonic if ρ(p,v) is positive for any p M and any v T M 0 . Plurisubharmonic
J p
L ∈ ∈ \ { }
functions play a very important role in almost complex geometry: they give attraction and localization
properties for pseudoholomorphic discs. For this reason the construction of J-plurisubharmonic functions
iscrucial.
Similarlytotheintegrable case, one maydefinethenotion ofpseudoconvexity inalmost complex mani-
folds. LetD beadomain in(M,J). Wedenote byTJ∂D := T∂D JT∂D theJ-invariant subbundle of
∩
T∂D.
Definition1.4.
(1) The domain D is J-pseudoconvex (resp. it strictly J-pseudoconvex) if ρ(p,v) 0 (resp. > 0)
J
L ≥
foranyp ∂D andv TJ∂D (resp. v TJ∂D 0 ).
∈ ∈ p ∈ p \{ }
(2) A J-pseudoconvex region is a domain D = ρ < 0 where ρ is a 2 defining function, J-
{ } C
plurisubharmonic onaneighborhood ofD.
Werecallthatadefiningfunction forD satisfiesdρ = 0on∂D.
6
WeneedthefollowinglemmaduetoE.Chirka[6].
Lemma 1.5. Let J be an almost complex structure of class 1 defined in the unit ball B of R2n satisfying
C
J(0) = J . ThenthereexistpositiveconstantsεandA = O(ε)suchthatthefunctionlog z 2+A z is
st ε ε
J-plurisubharmonic onBwhenever J J ε. k k k k
k − stk 1(B) ≤
C
SHARPESTIMATESOFTHEKOBAYASHIMETRICANDGROMOVHYPERBOLICITY 5
Proof. Thisisduetothefactthatforp Band J J sufficiently small,wehave:
∈ k − stk 1(B)
C
1 2
A z (p,v) A J(p) J
J st
L k k ≥ p − p k − k
(cid:16)k k k k
2(1+ J(p) J ) J J v 2
− k − stk k − stk 1(B) k k
C
(cid:17)
A
v 2
≥ 2 p k k
k k
and
2 1 2
ln z (p,v) J(p) J J(p) J 2 J J
LJ k k ≥ − p 2k − stk− p 2k − stk − p k − stk 1(B)
C
(cid:16) k k k k k k
2
J(p) J J J v 2
− p k − stkk − stk 1(B) k k
C
k k (cid:17)
6
J J v 2.
≥ − p k − stk 1(B)k k
C
k k
SotakingA= 24 J J theChirka’slemmafollows. (cid:3)
k − stk 1(B)
C
The strict J-pseudoconvexity of a relatively compact domain D implies that there is a constant C 1
≥
suchthat:
1
(1.5) v 2 ρ(p,v) C v 2,
J
Ck k ≤ L ≤ k k
forp ∂Dandv TJ(∂D).
∈ ∈ p
Let ρ be a defining function for D, J-plurisubharmonic on a neighborhood of D and strictly J-
plurisubharmonic on a neighborhood of the boundary ∂D. Consider the one-form dcρ defined by (1.2)
J
and let α be its restriction on the tangent bundle T∂D. It follows that TJ∂D = Kerα. Due to the strict
J-pseudoconvexity of ρ, the two-form ω := ddcρisa symplectic form (ie nondegenerate and closed) on a
J
neighborhood of∂D,thattamesJ. Thisimpliesthat
1
(1.6) g := (ω(.,J.)+ω(J.,.))
R
2
defines a Riemannian metric. We say that TJ∂D is a contact structure and α is contact form for TJ∂D.
Consequently vector fieldsinTJ∂D span thewhole tangent bundle T∂D. Indeed ifv TJ∂D, itfollows
∈
thatω(v,Jv) = α([v,Jv]) > 0andthus[v,Jv] T∂D TJ∂D. Wepointoutthatincasev TJ∂D,the
∈ \ ∈
vectorfieldsv andJv areorthogonal withrespecttotheRiemannianmetricg .
R
1.4. The Kobayashi pseudometric. The existence of local pseudoholomorphic discs proved by
A.Nijenhuis and W.Woolf [21] allows to define the Kobayashi-Royden pseudometric, abusively called the
6 FLORIANBERTRAND
Kobayashi pseudometric, K forp M andv T M:
(M,J) p
∈ ∈
1
K (p,v) := inf > 0,u : ∆ (M,J) J-holomorphic ,u(0) = p,d u(∂/∂x) = rv .
(M,J) 0
r →
n1 o
= inf > 0,u : ∆ (M,J), J-holomorphic ,u(0) = p,d u(∂/∂x) = v .
r 0
r →
Since the composition ofnpseudoholomorphic maps is still pseudoholomorphic, the Kobayashi psoeudo-
metricsatisfiesthedecreasing property:
Proposition 1.6. Let f : (M ,J ) (M,J) be a (J ,J)-holomorphic map. Then for any p M and
′ ′ ′ ′
→ ∈
v T M wehave
p ′
∈
K(M,J)(f(p),dpf (v)) K(M′,J′)(p,v).
≤
Sincethestructuresweconsideraresmoothenough,wemaydefinetheintegratedpseudodistance d
(M,J)
ofK :
(M,J)
1
d (p,q) := inf K (γ(t),γ˙ (t))dt, γ : [0,1] M, γ(0) = p,γ(1) = q .
(M,J) (M,J)
→
(cid:26)Z0 (cid:27)
Similarlytothestandard integrable case,B.Kruglikov[17]provedthattheintegrated pseudodistance ofthe
Kobayashi pseudometric coincides withtheKobayashi pseudodistance definedbychainsofpseudholomor-
phicdiscs.
WenowdefinetheKobayashihyperbolicity:
Definition1.7.
(1) Themanifold(M,J)isKobayashihyperboliciftheKobayashipseudodistance d isadistance.
(M,J)
(2) Themanifold (M,J) islocal Kobayashi hyperbolic atp M ifthere exist aneighborhood U ofp
∈
andapositiveconstantC suchthat
K (q,v) C v
(M,J)
≥ k k
foreveryq U andeveryv T M.
q
∈ ∈
(3) A Kobayashi hyperbolic manifold (M,J) is complete hyperbolic if it is complete for the distance
d .
(M,J)
2. GROMOV HYPERBOLICITY
In this section we give some backgrounds about Gromov hyperbolic spaces. Furthermore, according to
Z.M.BaloghandM.Bonk[1],proving thatadomain D withsomecurvature isGromovhyperbolic reduces
toproviding sharpestimatesfortheKobayashimetricK neartheboundary ofD.
(D,J)
2.1. Gromovhyperbolicspaces. Let(X,d)beametricspace.
Definition2.1. TheGromovproductoftwopointsx,y X withrespecttothebasepointω X isdefined
∈ ∈
by
1
(xy) := (d(x,ω) d(y,ω) d(x,y)).
ω
| 2 − −
TheGromovproduct measures the failure ofthe triangle inequality to bean equality and isalways non-
negative.
Definition2.2. ThemetricspaceX isGromovhyperbolic ifthereisanonnegative constant δ suchthatfor
anyx,y,z,ω X onehas:
∈
(2.1) (xy) min((xz) ,(z y) ) δ.
ω ω ω
| ≥ | | −
SHARPESTIMATESOFTHEKOBAYASHIMETRICANDGROMOVHYPERBOLICITY 7
Wepointoutthat(2.1)canalsobewrittenasfollows:
(2.2) d(x,y)+d(z,ω) max(d(x,z)+d(y,ω),d(x,ω)+d(y,z))+2δ,
≤
forx,y,z,ω X.
∈
ThereisafamilyofmetricspacesforwhichGromovhyperbolicitymaybedefinedbymeansofgeodesic
triangles. A metric space (X,d) is said to be geodesic space if any two points x,y X can be joined by a
∈
geodesic segment, thatistheimageofanisometryg :[0,d(x,y)] X withg(0) = xandg(d(x,y)) = y.
→
Such a segment is denoted by [x,y]. A geodesic triangle in X is the subset [x,y] [y,z] [z,x], where
∪ ∪
x,y,z X. Forageodesic space (X,d),onemaydefineequivalently (see[11])theGromovhyperbolicity
∈
asfollows:
Definition 2.3. The geodesic space X is Gromov hyperbolic if there is a nonnegative constant δ such that
foranygeodesic triangle [x,y] [y,z] [z,x]andanyω [x,y]onehas
∪ ∪ ∈
d(ω,[y,z] [z,x]) δ.
∪ ≤
2.2. Gromov hyperbolicity of strictly pseudoconvex domains in almost complex manifolds of dimen-
sionfour. LetD = ρ < 0 bearelatively compact J-strictly pseudoconvex smooth domaininanalmost
{ }
complex manifolds (M,J) of dimension four. Although the boundary of a compact complex manifold
with pseudoconvex boundary is always connected, this is not the case in almost complex setting. Indeed
D.McDuff obtained in [20] a compact almost complex manifold (M,J) of dimension four, with a discon-
nected J-pseudoconvex boundary. Since D is globally defined by a smooth function, J-plurisubharmonic
on a neighborhood of D and strictly J-plurisubharmonic on a neighborhood of the boundary ∂D, it fol-
lowsthatthe boundary ∂D ofD isconnected. Moreover this alsoimplies that there arenoJ-complex line
contained inDandsothat(D,d )isametricspace.
D,J
A 1 curve α : [0,1] ∂D is horizontal if α˙(s) TJ ∂D for every s [0,1]. This is equivalent to
C → ∈ α(s) ∈
α˙ 0. ThuswedefinetheLevilengthofahorizontal curveby
n
≡
1
1
Jρ length(α) := Jρ(α(s),α˙(s))2ds.
L − L
Z0
Wepointoutthat,dueto(1.6),
1
1
Jρ length(α) = gR(α(s),α˙(s))2ds.
L −
Z0
Since TJ∂D is a contact structure, a theorem due to Chow [7] states that any two points in ∂D may be
connected bya 1 horizontal curve. ThisallowstodefinetheCarnot-Carathe´odory metric asfollows:
C
d (p,q) := ρ length(α),α : [0,1] ∂D horizontal ,α(0) = p,α(1) = q .
H J
{L − → }
Equivalently,wemaydefinelocallytheCarnot-Carathe´odorymetric bymeansofvectorfieldsasfollows.
Consider two g -orthogonal vector fields v,Jv TJ∂D and the sub-Riemannian metric associated to
R
∈
v,Jv:
g (p,w) := inf a2+a2, a v(p)+a (Jv)(p) = w .
SR 1 2 1 2
Forahorizontal curveα,weset
(cid:8) (cid:9)
1
1
gSR length(α) := gSR(α(s),α˙(s))2ds.
−
Z0
8 FLORIANBERTRAND
Thuswedefine:
d (p,q) := g length(α),α :[0,1] ∂D horizontal ,α(0) = p,α(1) = q .
H SR
{ − → }
Wepointoutthatforasmallhorizontal curveα,wehave
α˙(s) = a (s)v(α(s))+a (s)J(α(s))v(α(s)).
1 2
Consequently
g (α(s),α˙(s)) = a2(s)+a2(s) g (α(s),v(α(s))).
R 1 2 R
AlthoughtheroleofthebundleTJ∂Discrucial,itisnotessentialtodefinetheCarnot-Carathe´odory metric
(cid:2) (cid:3)
with g instead of g . Actually, two Carnot-Carathe´odory metrics defined with different Riemannian
SR R
metricsarebi-Lipschitz equivalent (see[15]).
According to A.Bellaiche [2] and M.Gromov [15] and since T∂D is spanned by vector fields of TJ∂D
and Lie Brackets of vector fields of TJ∂D, balls with respect to the Carnot-Carathe´odory metric may be
anisotropically approximated. Moreprecisely
Proposition 2.4. Thereexistsapositive constantC suchthatforεsmallenough andp ∂D:
∈
ε
(2.3) Box p, B (p,ε) Box(p,Cε),
H
C ⊆ ⊆
whereB (p,ε) := q ∂D,d (p,q)(cid:16)<ε (cid:17)andBox(p,ε) := p+v ∂D, v < ε, v < ε2 .
H H t n
{ ∈ } { ∈ | | | | }
The splitting v = v +v is taken at p. We point out that choosing local coordinates such that p = 0,
t n
J(0) = J andTJ∂D = z = 0 ,thenBox(p,ε) = ∂D Q(0,ǫ),whereQ(0,ǫ)istheclassicalpolydisc
Q(0,ǫ) :=st z C02, z <{ε12, z }< ε . ∩
1 2
{ ∈ | | | | }
AsprovedbyZ.M.BaloghandM.Bonk[1],(2.3)allowstoapproximate theCarnot-Carathe´odory metric
byaRiemanniananisotropic metric:
Lemma2.5. Thereexistsapositive constantC suchthatforanypositiveκ
1
d (p,q) d (p,q) Cd (p,q),
κ H κ
C ≤ ≤
wheneverd (p,q) 1/κforp,q ∂D. Here,thedistanced (p,q)istakenwithrespecttotheRiemannian
H κ
≥ ∈
metricg definedby:
κ
g (p,v) := ρ(p,v )+κ2 v 2,
κ J h n
L | |
forp ∂Dandv = v +v T ∂D.
t n p
∈ ∈
Thecrucial ideaofZ.M.BaloghandM.Bonk[1]toprovetheGromovhyperbolicity ofD istointroduce
a function on D D, using the Carnot-Carathe´odory metric, which satisfies (2.1) and which is roughly
×
similartotheKobayashidistance.
Forp D wedefineaboundary projection mapπ : D ∂D by
∈ →
δ(p) = p π(p) = dist(p,∂D).
k − k
Wenoticethatπ(p)isuniquely determined onlyifp D issufficiently closetotheboundary. Weset
∈
1
h(p) := δ(p)2.
Thenwedefineamapg : D D [0,+ )by:
× → ∞
d (π(p),π(q))+max h(p),h(q)
H
g(p,q) := 2log { } ,
h(p)h(q) !
p
SHARPESTIMATESOFTHEKOBAYASHIMETRICANDGROMOVHYPERBOLICITY 9
for p,q D. The map π is uniquely determined only near the boundary. But an other choice of π gives a
∈
function g thatcoincides uptoabounded additiveconstant thatwillnotdisturb ourresults. Themotivation
of introducing the map g is related with the Gromov hyperbolic space Con(Z) defined by M.Bonk and
O.Schramm in [4] (see also [14]) as follows. Let (Z,d) be a bounded metric space which does not consist
ofasinglepointandset
Con(Z) := Z (0,diam(Z)].
×
Letusdefineamapg :Con(Z) Con(Z) [0,+ )by
× → ∞
d(z,z )+max h,h
′ ′
e g (z,h),(z′,h′) := 2log { } .
√hh
(cid:18) ′ (cid:19)
(cid:0) (cid:1)
M.BonkandO.Schrammin[4]provedthat(Con(Z),g)isaGromovhyperbolic (metric)space.
e
In our case the map g is not a metric on D since two different points p = q D may have the same
6 ∈
projection; nevertheless
e
Lemma2.6. Thefunctiong satisfies(2.2)(orequivalently (2.1))onD.
Proof. Letr berealnonnegativenumberssuchthat
ij
r = r and r r +r ,
ij ji ij ik kj
≤
fori,j,k = 1, ,4. Then
···
(2.4) r r 4max(r r ,r r ).
12 34 13 24 14 23
≤
1
Consider now four points pi D, i = 1, ,4. We set hi = δ(pi)2 and di,j = d(H,J)(π(pi),π(pj)).
∈ ···
Thenapplying(2.4)tor = d +min(h ,h ),weobtain:
ij i,j i j
(d +min(h ,h ))(d +max(h ,h ))
1,2 1 2 3,4 3 4
4max((d +max(h ,h ))(d +min(h ,h ),(d +min(h ,h ))(d +max(h ,h )).
1,3 1 3 2,4 2 4 1,4 1 4 2,3 2 3
≤
Then:
g(p ,p )+g(p ,p ) max(g(p ,p )+g(p ,p ),g(p ,p )+g(p ,p ))+2log4,
1 2 3 4 1 3 2 4 1 4 2 3
≤
whichprovesthedesiredstatement. (cid:3)
Asadirect corollary, ifa metric don D is roughly similar to g, then the metric space (D,d) isGromov
hyperbolic:
Corollary 2.7. LetdbeametriconD verifying
(2.5) C +g(p,q) d(p,q) g(p,q)+C
− ≤ ≤
for some positive constant C, and every p,q D. Then d satisfies (2.2) and so the metric space (D,d) is
∈
Gromovhyperbolic.
Z.M.Balogh and M.Bonk [1] proved that if the Kobayashi metric (with respect to J ) of a bounded
st
strictly pseudoconvex domain satisfies (0.1), then the Kobayashi distance is rough similar to the function
g. Theirproof ispurely metric anddoes notuse complex geometry or complex analysis. Wepoint out that
thestrictpseudoconvexity isonlyneeded toobtain(1.5)orthefactthatT∂D isspanned byvectorfieldsof
TJst∂D and Lie Brackets of vector fields of TJst∂D. In particular their proof remains valid in the almost
complexsetting and,consequently, TheoremAimplies:
10 FLORIANBERTRAND
Theorem2.8. LetDbearelativelycompactstrictlyJ-pseudoconvex smoothdomaininanalmostcomplex
manifold(M,J)ofdimension four. Thereisanonnegative constant C suchthatforanyp,q D
∈
g(p,q) C d (p,q) g(p,q)+C.
(D,J)
− ≤ ≤
AccordingtoCorollary2.7wefinallyobtainthefollowingtheorem(seealso(1)ofTheoremB):
Theorem2.9. LetDbearelativelycompactstrictlyJ-pseudoconvex smoothdomaininanalmostcomplex
manifolds(M,J)ofdimensionfour. Thenthemetricspace(D,d )isGromovhyperbolic.
(D,J)
Example2.10. Thereexist aneighborhood U ofp and adiffeomorphism z : U B R4, centered at p,
→ ⊆
such that the function z 2 is strictly J-plurisubharmonic on U and z (J) J λ . Hence the
st 2(U) 0
unitballBequipped wkithkthemetricd(B(0,1),z∗J) isGromovhyperbolikc.∗ − kC ≤
Asadirectcorollary ofExample2.10wehave(seealso(2)ofTheoremB):
Corollary 2.11. Let(M,J) beafour dimensional almost complex manifold. Thenevery point p M has
∈
abasisofGromovhyperbolic neighborhoods.
3. SHARP ESTIMATES OF THE KOBAYASHI METRIC
In this section we give a precise localization principle for the Kobayashi metric and we prove Theorem
A.
Let D = ρ < 0 be a domain in an almost complex manifold (M,J), where ρ is a smooth defining
{ }
strictlyJ-plurisubharmonic function. Forapointp Dwedefine
∈
(3.1) δ(p) := dist(p,∂D),
andforpsufficiently closeto∂D,wedefineπ(p) ∂Dastheuniqueboundary pointsuchthat:
∈
(3.2) δ(p) = p π(p) .
k − k
Forε> 0,weintroduce
(3.3) N := p D,δ(p) < ε .
ε
{ ∈ }
3.1. Sharp localization principle. F.Forstneric and J.-P.Rosay [9] obtained a sharp localization principle
of the Kobayashi metric near a strictly J -pseudoconvex boundary point of a domain D Cn. However
st
⊂
their approach is based on the existence of some holomorphic peak function at such a point; this is purely
complex and cannot be generalized in the nonintegrable case. The sharp localization principle we give is
basedonsomeestimatesoftheKobayashi lengthofapathneartheboundary.
Proposition 3.1. There exists a positive constant r such that for every p D sufficiently close to the
∈
boundary and for every sufficiently small neighborhood U of π(p) there is a positive constant c such that
foreveryv T M:
p
∈
(3.4) K (p,v) (1 cδ(p)r)K (p,v).
(D U,J) (D U,J)
∩ ≥ − ∩
We will give later a more precise version of Proposition 3.1, where the constants c and r are given
explicitly (seeLemma3.4).
Proof. Weconsideralocaldiffeomorphism z centered atπ(p)fromasufficiently smallneighborhood U of
π(p)toz(U)suchthat
(1) z(p) = (δ(p),0),
(2) thestructure z J satisfiesz J(0) = J andisdiagonal,
st
∗ ∗