Behavior of the Bergman kernel and metric near convex boundary points

7 Pages·0.12 MB·English

by  Nikolai Nikolov

BEHAVIOR OF THE BERGMAN KERNEL AND
METRIC NEAR CONVEX BOUNDARY POINTS
2
0 NIKOLAI NIKOLOV AND PETER PFLUG
0
2
Abstract. The boundary behavior of the Bergman metric near

n a convex boundary point z0 of a pseudoconvex domain D ⊂ Cn
a is studied; it turns out that the Bergman metric at points z ∈ D
J
0 in direction of a fixed vector X0 ∈ Cn tends to infinite, when z is
1 approaching z0, if and only if the boundary of D does not contain
any analytic disc through z0 in direction of X0.

]
V
C For a domain D ⊂ Cn we denote by L2(D) the Hilbert space of all
. h
h holomorphic functions f that are square-integrable and by ||f|| the
D
t
a L -norm of f. Let K (z) be the restriction on the diagonal of the
2 D
m
Bergman kernel function of D. It is well-known (cf. [5]) that
[
K (z) = sup{|f(z)|2 : f ∈ L2(D), ||f|| ≤ 1}.
D h D
1
v IfK (z) > 0forsomepointz ∈ D,thentheBergmanmetricB (z;X),
D D
8 X ∈ Cn, is well-defined and can be given by the equality
8
0 M (z;X)
D
1 B (z;X) = ,
D
0 K (z)
D
2
0 where p
/
h M (z;X) = sup{|f′(z)X| : f ∈ L2(D), ||f|| = 1, f(z) = 0}.
t D h D
a We say that a boundary point z of a domain D ⊂ Cn is convex if
m 0
there is a neighborhood U of this point such that D ∩U is convex.
:
v
In [4], Herbort proved the following
i
X
Theorem 1. Let z be a convex boundary point of a bounded pseudo-
r 0
a convex domain D ⊂ Cn whose boundary contains no nontrivial germ
of an analytic curve near z . Then
0
lim B (z;X) = ∞
D
z→z0
for any X ∈ Cn \{0}.
Herbort’s proof is mainly based on Ohsawa’s ∂¯-technique. The main
purpose of this note is to generalize Theorem 1 using more elementary
methods.
2000 Mathematics Subject Classification. Primary: 32A25.
Key words and phrases. Bergman kernel, Bergman metric.
1
2 NIKOLAI NIKOLOV AND PETER PFLUG
For a convex boundary point z of a domain D ⊂ Cn we denote by
0
L(z ) the set of all X ∈ Cn for which there exists a number ε > 0
0 X
such that z + λX ∈ ∂D for all complex numbers λ, |λ| ≤ ε . Note
0 X
that L(z ) is a complex linear space.
0
Then our result is the following one.
Theorem 2. Let z be a convex boundary point of a bounded pseudo-
0
convex domain D ⊂ Cn and let X ∈ Cn. Then
(a) liminfK (z)dist2(z,∂D) ∈ (0,∞];
D
z→z0
(b) lim B (z;X) = ∞ if and only if X 6∈ L(z ). Moreover, this
D 0
z→z0
limit is locally uniform in X 6∈ L(z );
0
(c) If L(z ) = {0}, then (a) and (b) are still true without the as-
0
sumption that D is bounded.
Proof of Theorem 2. Toprove(a)and(b)wewillusethefollowing
localization theorem for the Bergman kernel and metric [2].
Theorem 3. Let D ⊂ Cn be a bounded pseudoconvex domain and let
V ⊂⊂ U be open neighborhoods of a point z ∈ ∂D Then there exists a
0
constant C˜ ≥ 1 such that
C˜−1K (z) ≤ K (z) ≤ K (z),
D∩U D D∩U
C˜−1B (z;X) ≤ B (z;X) ≤ C˜B (z;X)
D∩U D D∩U
for any z ∈ D ∩ V and any X ∈ Cn. (Here K (z) and B (z;·)
D∩U D∩U
denote the Bergman kernel and metric of the connected component of
D∩U that contains z.)
So, we may assume that D is convex.
To prove part (a) of Theorem 2, for any z ∈ D we choose a point
z˜∈ ∂D such that ||z −z˜|| = dist(z,∂D). We denote by l the complex
line through z and z˜. By the Oshawa-Takegoshi extension theorem for
L2–holomorphic functions [7], it follows that there exists a constant
C > 0 only depending on the diameter of D (not on l) such that
1
(1) K (z) ≥ C K (z).
D 1 D∩l
Since D ∩l is convex, it is contained in an open half-plane Π of the
l-plane with z˜∈ ∂Π. Then
1
(2) K (z) ≥ K (z) = .
D∩l Π 4πdist2(z,∂Π)
Now, part (a) of Theorem 2 follows from the inequalities (1), (2) and
the fact that dist(z,∂Π) ≤ ||z −z˜|| = dist(z,∂D).
To prove part (b) of Theorem 2, we denote by N(z ) the complex
0
affine space through z that is orthogonal to L(z ). Set E(z ) = D ∩
0 0 0
3
N(z ). Note that E(z ) is a nonempty convex set. So, part (b) of
0 0
Theorem 2 will be a consequence of the following
Theorem 4. Let z be a boundary point of a bounded convex domain
0
D ⊂ Cn. Then:
(i) lim B (z;X) = ∞ locally uniformly in X 6∈ L(z );
D 0
z→z0
(ii) for any compact set K ⊂⊂ E(z ) there exists a constant C > 0
0
such that
B (z;X) ≤ C||X||, z ∈ K0, X ∈ L(z ),
D 0
where K0 := {z +tz : z ∈ K, 0 < t ≤ 1} is the cone generated by K.
0
Proof of Theorem 4. To prove (i) we will use the well-known fact
thattheCarath´eodorymetricC (z;X)ofD doesnotexceedB (z;X).
D D
On the other hand, we have the following simple geometric inequality
[1]:
1
C (z;X) ≥ ,
D
2d(z;X)
where d(z;X) denotes the distance from z to the boundary of D in the
X-direction, i.e. d(z;X) := sup{r : z +λX ∈ D, λ ∈ C, |λ| < r}. So,
if we assume that (i) does not hold, then we may find a number a > 0
and sequences D ⊃ (z ) , z → z , Cn ⊃ (X ) , X → X 6∈ L(z )
j j j 0 j j j 0
1
such that B (z ;X ) ≤ . Hence d(z ;X ) ≥ a which implies that
D j j j j
2a
for |λ| ≤ a the points z +λX belong to D¯ and, in view of convexity,
0
they belong to ∂D. This means that X ∈ L(z ) – contradiction.
0
To prove part (ii) of Theorem 4, we may assume that z = 0 and
0
L := L(0) = {z ∈ Cn : z = ... = z = 0} for some k < n. Then
1 k
N := N(0) = {z ∈ Cn : z = ... = z = 0}. From now on we will
k+1 n
write any point z ∈ Cn in the form z = (z′,z′′), z′ ∈ Ck, z′′ ∈ Cn−k.
Note that L ∈ ∂D near 0, i.e. there exists an c > 0 such that
(3) {0′}×∆′′ ∈ ∂D,
c
where ∆′′ ⊂ Cn−k is the polydisc with center at the origin and radius
c
c. Since K ⊂⊂ E := E(0) and since E is convex, there exists an α > 1
such that K ⊂⊂ E , where E := {z : αz ∈ E}. Note that K0 ⊂ E .
α α α
Using (3), the following equality
1 1 1
(z′,z′′) = (αz′,0′′)+(1− )(0′,(1− )−1z′′),
α α α
and the convexity of D, it follows that
(4) F ×∆′′ ⊂ D,
α ε
4 NIKOLAI NIKOLOV AND PETER PFLUG
1
where ε := c(1 − ) and where F is the projection of E in Ck (we
α α
α
can identify E with F ). For δ := c(α − 1) we get in the same way
α α
that
(5) D˜ := D ∩(Ck ×∆′δ′) ⊂ F1 ×∆′δ′.
α
Now, let (z,X) ∈ K0 ×L. Note that z = (z′,0′′) and X = (0′,X′′).
Then, using (4) and the product properties of the Bergman kernel and
metric, we have
(6) MD(z;X) ≤ MFα×∆′ε′(z;X) =
= M∆′ε′(0′′;X′′) KFα(z′) ≤ C1||X|| KFα(z′)
for some constant C1 > 0. pOn the other side, spince K0 ⊂⊂ Ck ×∆′δ′,
in virtue of Theorem 3 there exists a constant C˜ ≥ 1 such that
K (z) ≥ C˜−1K (z).
D D˜
Moreover, in view of (5), we have
KD˜(z) ≥ KF1(z′)K∆′′(0′′)
δ
α
and hence
(7) K (z) ≥ (C )2K (z′)
D 2 F1
α
for some constant C > 0. Now, by (6) and (7), it follows that
2
M (z;X) C K (z′)
(8) B (z;X) = D ≤ 1||X|| Fα .
D
KD(z) C2 sKF1(z′)
α
Note that z′ → α−2z′ is pa biholomorphic mapping from F1 onto Fα
α
and, therefore,
(9) K (z′) = α−4kK (α−2z′).
F1 Fα
α
In view of (8) and (9), in order to finish (ii) we have to find a constant
C > 0 such that
3
(10) K (z′) ≤ C K (α−2z′)
Fα 3 Fα
for any z′ ∈ H0 with H0 := {tz′ : z′ ∈ H, 0 < t ≤ 1}, where H is the
projection of K into Ck (we can identify K with H).
To do this, note first that γ := dist(H,∂F ) > 0 since K ⊂⊂ E .
α α
Fix τ ∈ (0,1] and z′ ∈ H0, and denote by Tτ,z′ the translation that
maps the origin in the point τz′. It is easy to check that
(11) Tτ,z′(F¯α ∩Bγ) ⊂ Fα,
5
where B is the ball in Ck with center at the origin and radius γ. To
γ
prove (10), we will consider the following two cases:
Case I. z′ ∈ H0 \Bγ ⊂⊂ Fα: Then
2
m
(12) K (z′) ≤ 1K (α−2z′),
Fα m Fα
2
where m := sup K and m := infK .
1 Fα 2 Fα
H0\Bγ
2
Case II. z′ ∈ H0∩Bγ: By Theorem 3 there exists a constant C˜3 ≥ 1
2
such that C˜3KFα ≥ KFα∩Bγ on Fα ∩Bγ2. In particular,
(13) C˜ K (α−2z′) ≥ K (α−2z′).
3 Fα Fα∩Bγ
On the other side, by (11) with data T := T1−α−2,z′ it follows that
(14) K (α−2z′) = K (z′) ≥ K (z′).
Fα∩Bγ T(Fα∩Bγ) Fα
m
Now,(12),(13),and(14)implythat(10)holdsforC := max{ 1,C˜ }.
3 m 3
2
This completes the proofs of Theorem 4 and part (b) of Theorem 2.
Remark. The approximation (5) of the domain D ∩(Ck ×∆′′) by
δ
the domain E1 ×∆′δ′ can be replaced by using the Oshawa-Takegoshi
α
theorem [7] with the data D and N.
Finally, part (c) of Theorem 2 will be a consequence of the following
two theorems.
Theorem 5. [6] Let D ⊂ Cn. be a pseudoconvex domain and let U
be an open neighborhood of a local (holomorphic) peak point z ∈ ∂D.
0
Then
K (z)
D
lim = 1
z→z0 KD∩U(z)
and
B (z;X)
D
lim = 1
z→z0 BD∩U(z;X)
locally uniformly X ∈ Cn \{0}.
Theorem 6. Let z be a boundary point of a bounded convex domain
0
D ⊂ Cn. Then the following conditions are equivalent:
1. z is a (holomorphic) peak point;
0
2. z is the unique analytic curve in D¯ containing z ;
0 0
3. L(z ) = {0}.
0
Notethattheonlynontrivialimplicationis(3) =⇒ (1)itiscontained
in [8]. Now, part (c) of Theorem 2 is a consequence of this implication,
Theorem 5, and part (b) of Theorem 2.
6 NIKOLAI NIKOLOV AND PETER PFLUG
Proof of Theorem 6. The implication (2) =⇒ (3) is trivial.
The implication (1) =⇒ (2) easily follows by the maximum principle
andthefactthat therearea neighborhoodU ofz andavector X ∈ Cn
0
such that (D¯ ∩U)+(0,1]X ⊂ D (cf. (11)).
Denote by A0(D) the algebra of holomorphic functions on D which
are continuous on D¯. Now, following [8] we shall prove the implication
(3) =⇒ (1); namely, (3) implies that z is a peak point with respect to
0
A0(D). This is equivalent to the fact (cf. [3]) that the point mass at
z is the unique element of the set A(z ) of all representing measures
0 0
for z with respect to A0(D), i.e. supp µ = {z } for any µ ∈ A(z ).
0 0 0
Let µ ∈ A(z ). Since D is convex, we may assume that z = 0 and
0 0
D ⊂ {z ∈ Cn : Re(z ) < 0}. Note that if a is a positive number
1
such that a inf Re(z ) > −1 (D is bounded), then the function f (z) =
1 1
z∈D
exp(z +az2) belongs to A0(D) and |f (z)| < 1 for z ∈ D¯ \{z = 0}.
1 1 1 1
This easily implies (cf. [3]) that supp µ ⊂ D := ∂D∩{z = 0}. Since
1 1
L(0) = 0, the origin is a boundary point of the compact convex set
D . As above, we may assume that D ⊂ {z ∈ Cn : Re(z ) ≤ 0} (z is
1 1 2 2
independent of z ) and then construct a function f ∈ A0(D) such that
1 2
|f (z)| < 1forz ∈ D \{z = 0}. Thisimpliesthatsupp µ ⊂ D ∩{z =
2 1 2 1 2
0}. Repeating this argument we conclude that supp µ = {0}, which
completes the proofs of Theorems 6 and 2.
Acknowledgments. This paper was written during the stay of the
first author at the University of Oldenburg supported by a grant from
the DAAD. He likes to thank both institutions, the DAAD and the
Mathematical Department of the University of Oldenburg.
References
[1] E. Bedford, S. Pinchuk, Convex domains with noncompact automorphism
groups, Sb. Math. 82 (1995), 3-26.
[2] K. Diederich, J.E. Fornaess, G. Herbort, Boundary behavior of the Bergman
metric, Proc. Symp. Pure Math. 41 (1984), 59-67.
[3] T.W. Gamelin, Uniform algebras, Chelsea, New York, 1984.
[4] G. Herbort, On the Bergman metric near a plurisubharmonic barrier point,
Prog. Math. 188 (2000), 123-132.
[5] M.Jarnicki,P.Pflug,Invariant distances and metrics in complex analysis, Wal-
ter De Gruyter, Berlin, New York, 1993.
[6] N. Nikolov, Localization of invariant metrics, Arch. Math. (to appear).
[7] T.Ohsawa,K.Takegoshi,Extension of L2 holomorphic functions,Math.Z.195
(1987), 197-204.
[8] N. Sibony, Une classe de domaines pseudoconvexes, Duke Math. J. 55 (1987),
299-319.
7
Institute of Mathematics and Informatics, Bulgarian Academy of
Sciences, 1113 Sofia, Bulgaria
E-mail address: [email protected]
CarlvonOssietzkyUniversita¨tOldenburg,FachbereichMathematik,
Postfach 2503, D-26111 Oldenburg,Germany
E-mail address: [email protected]