PROCEEDINGSof the
AMERICANMATHEMATICAL
SOCIETY
Volume 122, Number 4, December 1994
SOME ESTIMATES OF THE KOBAYASHIMETRIC
IN THE NORMALDIRECTION
SIQIFU
(Communicated by Eric Bedford)
Abstract. In this paper, we study the behavior of the Kobayashi metric in
the normal direction near a Levi-pseudoconvex boundary point of a smoothly
bounded domain without assuming global pseudoconvexity. As a corollary, we
obtain a characterization of pseudoconvexity by the rate of the growth of the
Kobayashi metric in the normal direction.
I. Introduction
and theorems
Let ficcC"
be a bounded domain with smooth boundary near z0 e dil.
Let U be a neighborhood of z0. Let r{z) be a local defining function of fí
on U, i.e.,
CinU = {zeU
and riz) e &°°(U),
Vr(z)\aanu
| r(z)<0}
s idriz)/dzx,
driz)/dz2,...
, driz)/dzn)
¿
0.
Let A be the unit disc and let A,, = {yÇ;
of Q is defined by
Ç e A} . The Kobayashi metric
Faiz, X) = inf {1¡X \ 3f:A-+Yl is holomorphic and
/(0) = z,/'(0) = AX,a>0}.
For z = (zi, z2, ... , z„) 6 C", set z' = (zx, z2,... , z„_i). Let d(z) =
dist(z, öQ) and let 7t(z) be the projection to the boundary for z near z0
such that diz) = \z- n(z)\. Let Nn(Z) be the inward normal direction at 7t(z).
Denote
HZ0= jX e C" | (dr(zo),X) = J2 ]jy.(zo)Xi= o| »
We call z0 edü,
a Levi-pseudoconvex point if
',7=1
'
Received by the editors March 24, 1993.
1991 MathematicsSubject Classification.Primary 32H15.
Key words and phrases. Kobayashi metric, pseudoconvex domain.
©1994 American Mathematical Society
0002-9939/94 $1.00+ $.25 perpage
1163
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1164
SIQIFU
In [K], Krantz has proven that if Í2 is a bounded domain with smooth boundary
near z0, then there exists constant C > 0 such that
Fa(z, Vr(n(z)))> ClVff™
for z e Q near z0.
In this note, we shall prove the following theorems.
Theorem A. Let Í2 ce C" be a bounded domain with smooth boundary near
z0 e dYl. Suppose that there exist a > 3/4, C > 0, X e C" \ HZo, and
{zk}h=x with zk -* zo nontangentially (i.e., zk stays in some cone A with
vertex at zo and axis NZo) such that
(l.i)
F^zk>x^c]{dT»?z'k)X)l
foral1 keK
Then zo is a Levi-pseudoconvex point.
Theorem B. Let QccC"
be a bounded domain with smooth boundary near
zo e dYl, and let A be a cone with vertex at zo and axis JVZo.Assume that
zo is the origin. Suppose that the local defining function of Yl has the following
form near z0 :
r(z) = Rez„+^(|z'r
+ |zn|.|z|).
Then there exist a neighborhood V of zo and a constant C > 0 such that
(1-2)
Fa(z,X)>C-^-
dX~m(z)
for z e A n fí n V and all X eCn . Furthermore, there exists Cx > 0 such that
(1.3)
for zeAnflnF
Fn(z,X)>Cx
\*n\
dx~ù(z)
and X e Cn with \X\ < K\X„\ (Cx may depend on the
constant K).
As corollaries of Theorem A and Theorem B, we obtain
Corollary 1. Lei QccC"
be a bounded domain with smooth boundary near
z0 e dYl, and let A be a cone with vertex at z0 and axis NZo. Then z0 is
a Levi-pseudoconvex point if and only if there exist a neighborhood V of z0,
a > 3/4, and C > 0 such that
(1.4)
Ffl(z,Vr(z0))>ci|M
for all z e A n fí n V.
Corollary 2. Let flccC2
be a bounded domain with smooth pseudoconvex
boundary near zq e d£l and let A be a cone with vertex at zo and axis NZo.
Then for each a e (0, 1), there exist a neighborhood V of z0 and a constant
C > 0 such that
(1.5)
for all zeAnYinV
FQiz,X)>C^^)\
and X eCn.
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SOME ESTIMATESOF THE KOBAYASHIMETRIC IN THE NORMAL DIRECTION
1165
II. Proofs of the theorems
Some ideas in the proofs of Theorem A and Theorem B come from [K]. We
will also need the following lemma, which can be proven directly from Lemma
2 in [R].
Lemma. Let Yl' be a subdomain of a bounded domain Q with dYl' n dYl D
UndCl for some neighborhood U of z0edYl. Then there exist a neighborhood
V cc U of zo and a constant C > 0 such that
(2.1)
for zeYl'nV
Faiz,X)<CFQiz,X)
and XeC.
We will use C to denote constants which may be different in different appearances.
Proof of Theorem A. After a translation and a unitary transformation, we may
assume that zo is the origin and dYl is locally defined by
n
(2.2)
riz) = Re z„ + £ üíjZíZj + cf (|z|3)
.»;=i
for z near zq .
Suppose that dYl is not Levi-pseudoconvex at z0. Then the matrix
J2„
(d2rjzo)\
\dz¡dZj)x<i l<i,j<n-l
has at least one negative eigenvalue . Therefore, after a unitary transformation
in z' = {z\, Zi,... , zH-\) and a simple change of coordinate system, we may
assume that
n
riz) = Rez„ - \zx\2 + £
a,7z,z,- + ¿f(|z|3)
i,j=2
for z e U, where U is some neighborhood of zq . Shrinking U, we have
(2.3)
r(z)<Rez„-&
I
|2
2
"
+ C]T|z,-|2,
i=2
for
zeU.
LeXA = {- Re z„ > k\z\) ( 0 < k < 1 ) be the cone. By the Implicit Function
Theorem,
(2.4)
lim
^1^
z—o diz)
= 1-
zeAnn
By the homogeneity of the Kobayashi metric, we may assume that X =
iXx,X2,...
,X„_i, 1). For z = iz', zn) = izx,z2,...
, zn) e Anfin U, let
S = -Rez„. Define *¿(f) - (*W(C),«2i(0, ••• >*«¿(0) by
r53/4
01<5= z, + ^-ZiC
¿3/4
*w = z* +-j-^fcC,
+ 2C2;
for 2</c</i-l;
¿3/4
<I>«¿
= Zn+ —C
Then4>,(0)= z, <DJ(0)
= Çl.
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1166
SIQIFU
Claim. There exists y e (0, 1) such that for all S > 0 sufficiently small, we
have Os(Ay) cYlnlf.
Proof of the Claim. By choosing y e (0, 1) small enough, we have O^A,,) c
U. It follows from (2.3) that
r(*¿(0) < S + ^Ç- ReC- |a>1¿2(°|2
+ C ¿ |a>1(5(C)|2.
i=2
Since 6 > k\z\, we see that when r5 is sufficiently small,
(2.5,
r(*j(0)<_«+ÇReC_M
For Kl < ^1/4, by (2.5),
r(o,(C))<4+ç.^_M
à
4
|*m(C)|2 ,n
2
For ICI> <51/4
, we have
¿3/4
|Ol,5(C)|2>2|C|4-|z,-r-^-^C|2
> 2|C|4 - Cr53/2.
Thus (2.5) implies that when S sufficiently small,
<_-»+<^+iäl_lfl«<0.
This concludes the proof of the Claim.
Next, by the Claim and the definition of the Kobayashi metric,
C
Fanu(z, X) < -pjj.
Thus, combining (2.4) and the length-decreasing property of Kobayashi metric,
we obtain
C
Fa{z'X)-dyW)'
which contradicts (1.1). Therefore, dQ is Levi-pseudoconvex at z0 . D
Proof of Theorem B. By the assumption, there exists a neighborhood U of zn
such that
(2.6)
Ííni7c{zei7|
Rez„-C(|z'r
+ |z„|-|z|)<0}.
Let A = {-Rez„ > k\z\) (k e (0, 1)). For z e Anílní/
and X e C"
(by homogeneity of the Kobayashi metric, we may assume that |A"| < 1 ), let
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SOMEESTIMATESOF THE KOBAYASHIMETRICIN THE NORMALDIRECTION
1167
O(C)= (0(C), O„(0) = (<D,(C),
O2(0, ... , *B(C)):A - ClnU be an analytic
disc satisfying
(2.7)
<D(0)= z,
<t>'iO)
= XX,
where X > 0 is a constant to be estimated. By the Cauchy Integral Formula, we
have
(2.8)
\<J>iiO-Zi\<C\Q,
l<i<n,
and
(2.8')
10,(0 - Zi- XXiQ< C|C|2,
1 < i < n,
for Id < 1/2. Also, by (2.6), we have
(2.9)
Reo„(c)
<c (i*(or+i*»(oi•mo\)■
Denote Ô = - Rez„ . By (2.8), ( 2.8' ), and the fact that k\z\ < a, we have
|O(0|<c(|z| + |C|)
(2.10)
<c(il/k)ô
<£.l/2(5l/m>
+ cÔx'm)
fof
\Q<côl/m
and
|<D(C)|<c(|z| + (A|X|)|C|+ |C|2)
(2.10')
< C (il/k)ô + iX\X\)cöxl2m+ c2ôx'm}
< ci/2 (¿i/m + (/i|X|)r51/2m),
for ICI< C(51/2m
when c, r5 are sufficiently small.
Now, it follows from (2.10), ( 2.10' ), and (2.9) that
(2.11)
Re<P„(C)<^ + ^|<Dn(C)|. for |f| < c8x'm
and
(2.11')
Re<Dn(C)<¿ + (^l)m<5l/2 + ^|On(C)|,
for \Q<côx'2m
when c, S are sufficiently small.
Denote
Dió= {WeC
| ReW<S + ^f)mSl/2
+ l-\W\},
for , = 1, 2
where ex = 0, e2 = 1. LeX gx(Q = <t>n(côxlmQand g2(Q = <Î>„(côxl2mQ.
By (2.7), (2.11), and ( 2.11' ), we have gi(A) C DiS, gi(0) = zn ( i = 1, 2 ),
íí(0) = kXncôxlm , and g'2(0) = XXncôxl2m. However, it is clear that
DiScDiS = £\^we£
| Imu; = 0,Rew>¿
Thus g¡(A) c DiS.
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+ e,(A|*|)m(51'/2}.
1168
SIQIFU
Since
C
\^,1)>
ô+Bt{x\x\y*ôw
where C > 0 is a constant independent on <5,it follows that
\g¡iO)\<Cio + eiiX\X\rox'2).
Therefore,
(2.12)
and
(2.13)
X\Xn\ôl/m< Co
X\Xn\ôxl2m
< Ció + iX\X\)môxl2).
Now by (2.12), X\Xn\< Côx~xlm. Thus (1.2) is valid. Furthermore, when
|Ar| < K\Xn\, it follows from (2.13) that X\Xn\< Côx~x/2m where C > 0 may
depend on K. Therefore, (1.3) follows. D
Proof of Corollary 1. The sufficiency comes directly from Theorem A. We
now prove the necessity. Suppose that zo is a Levi-pseudoconvex point. After
a change of coordinates, we may assume that zq is the origin and dYl is locally
defined by
n
riz) = Re zn + £
a¡jZiZj + c?i\z\3)
',7=1
near z0.
Since the matrix (a¡j){<i ,<„_i is positive semidefinite,
r(z)>Rez„
+ 2Re Ij^fl»;^^/)
= Rez„+c?i\z'\i
+ann\zn\2+ c?i\z\3)
+ \zn\'\z\).
Therefore, (1.4) is valid for a = 5/6 by Theorem B. D
Proof of Corollary 2. When zo is of finite type, as in Property 2.1 in [M],
we can construct a smoothly bounded pseudoconvex domain Q' c Yl such that
dYl' n dYl D dYl n U for some neighborhood U of zq . By the Lemma, there
exist a neighborhood V of zo and a constant C > 0 such that
Fa{z,X)>CFaiz,X)
for z e V n Yl', X e C" . Applying Theorem 1 in [C] to fi', we then obtain
that (1.5) is valid for a = 1.
When zo is of infinite type, then for each m e N, after a change of coordinates, the local defining function of Yl near z0 has the form
riz) = Rez2 + cfi\zx\m + \z2\'\z\).
Thus, (1.5) follows from Theorem B. G
Remark. The estimates in Theorem B are sharp. Suppose that Yl is a bounded
domain such that ôQ is locally defined by r(z) = Rez2-|zi|2m near the origin.
Then for Ps = {0,-3), X = iô-x+x'2m, 1), and Y = (0, 1), we have
*(*•*) **I=W' F°^'r>*¿T=W
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SOME ESTIMATESOF THE KOBAYASHIMETRIC IN THE NORMAL DIRECTION
1169
Acknowledgment
The author thanks Professor Steven Krantz for his advice and encouragement.
References
[C]
D. Catlin, Estimates of invariant metrics on pseudoconvex domains of dimension two, Math.
Z. 200 (1989),429-466.
[K]
S. Krantz, The boundary behavior of the Kobayashi metric, Rocky Mountain J. Math. 22
[M]
J. McNeal, Lower bounds on Bergman metric near a point of finite type, Ann. of Math. (2)
[R]
H. Royden, Remarks on the Kobayashi metric, Several Complex Variables II, Lecture Notes
(1992).
136 (1992), 339-360.
in Math., vol 185, Springer,New York, 1971,pp. 125-137.
Department
of Mathematics,
Washington University, St. Louis, Missouri 63130
E-mail address : s f uQmat h. wnst 1. edu
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