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ANNALES DE L’INSTITUT FOURIER
KARL OELJEKLAUS
PETER PFLUG
EL HASSAN YOUSSFI
The Bergman kernel of the minimal ball
and applications
Annales de l’institut Fourier, tome 47, no 3 (1997), p. 915-928
<http://www.numdam.org/item?id=AIF_1997__47_3_915_0>
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http://www.numdam.org/
Ann. Inst. Fourier, Grenoble
47, 3 (1997), 915-928
THE BERGMAN KERNEL OF THE
MINIMAL BALL AND APPLICATIONS
by K. OELJEKLAUS, P. PFLUG & E.H. YOUSSFI
1. Introduction.
Let B* be the domain in C71, n > 2, defined by
B* ^ { ^ e C 7 1 : l^+l^^l < l},
n ^ p )
where z • z := ^ zj. This is the ball of radius — with respect to the
j=i 2
norm
/ \ y 2 I | y A y|
N ^ ' ^ d ^ +p z ' , ^ec".
V ^
The norm A^ was introduced by Hahn and Pflug [HP], and was shown to
be the smallest norm in C71 that extends the euclidean norm in M71 under
certain restrictions. The automorphism group of B^ is compact and its
identity component is Aut^(B^) = S1 ' SO{n,R), where the S^-action is
diagonal and the 50 (n, Reaction is the matrix multiplication, see [K] or
[OY]. This shows that for n > 3, the ball B^ is not biholomorphic to any
Reinhardt domain. For n = 2, B^ is linearly biholomorphic to the Reinhardt
triangle {(^1,^2) € C2 : \z^\ + \z^\ < 1}.
The main purpose of this note is to establish the following
Key words: Bergman kernel - Minimal ball - Proper holomorphic mapping.
Math. classification: 32H10 - 32H35.
916 K. OELJEKLAUS, P. PFLUG & E.H. YOUSSFI
THEOREM. — The Berg-man kernel ofB^c is given by the formula
KB, (^ w)
i E% (2^i)xn-l~2JyJ (2nx ~{n~mx2 ~y])
= n ( n + i ) y ( B , ) ( x 2 - r ) ^ 1 '
where
X = l — <2;,w>, and V = (^ • z)w • w,
and V(B^) is the Lebesgue volume orB^.
In particular, when n = 2, the Bergman kernel of B^ is
K^ (^ w)
2 3(1- < z, w ^(l^- < z, w >) + (z • z)ww(5 - 3 < z, w >)
7T2 ((1- < Z,W >)2- (^•^)W~W)3
It should be noted that for n == 2 this formula can be obtained from the
Bergman kernel of the above mentioned Reinhardt triangle whose Bergman
kernel can be found in ([JP], p. 176).
Remark. — To the best of our knowledge, the domain B+ is the first
bounded domain in C71 which is neither Reinhardt nor homogeneous, and
for which we have an explicit formula for its Bergman kernel.
2. Preparatory results.
Let G be a semi-simple complex Lie group and K a maximal compact
subgroup of G. Suppose that G acts irreducibly on a finite dimensional
complex vector space E\ via a representation (IIA,£'A) with dominant
weight A and dominant vector v\. Assume further that EA is furnished
with a .?f-invariant hermitian scalar product [•, '] . If G = KAN is the
Iwasawa decomposition of G, let Q denote the sum of the roots associated
with the complex decomposition in the Lie algebra Q of G. If a is the Lie
algebra of A, then we have the following orthogonal decomposition with
respect to the Cartan-Killing form
a== aACa^:,
where OA is the annihilator of A. If HQ is the unique vector in a^- such that
A(J^o) = 1, we set
(2.1) a := 2g(Ho).
THE BERGMAN KERNEL OF THE MINIMAL BALL 917
Let M* be the intersection of the G-orbit of VA and the unit ball in E^.
In his work [Lo], Loeb proved that the Bergman kernel of the manifold
M* with an invariant form on M* is given for ^ == ^A(gi)vA^ =
^A(92)vA,gi,g2 € G, by
00
(2.2) ^M-(C^) =^(2j+a)TAa)[C^
j=o
where T^(j) is the dimension of the representation with dominant weight
J'A.
Here we consider the special case G = S0(n + 1,C) with its natural
linear representation on the complex hermitian space (C7^1^ • , • >),
where A is the dominant weight associated to this representation and
VA = ""^(^z? 0, • • • , 0). The intersection of the G-orbit of v\ and the unit
ball in C^ is M* = M \ {0}, where
M'={z=(z^"•,Zn^)eCn+l: \z\ <l,z^z=0}.
Then by formula (2.2), we see that the Bergman kernel of M* with respect
to an S0(n-\-l, (C)-invariant form a(z)/\a{z) is given (up to a multiplicative
constant) by
00
(2.3) ^M-(C^) = ̂ (2j^-aWj) < C^ >3,
3=0
for C,^ e M*.
LEMMA 2.1. — If' a{z} is an S0{n-\- 1,C)-invariant nonzero n-form
on M* (the invariant n-from a is unique up to a constant), then the
Bergman kernel of M* with respect to the invariant form a(z) A a(z) is
given (up to a multiplicative constant) by
2(n+l)<C,y;>+2n-2
^(c^ (1-<^»^
_ ____4n____ 2n+2
~ (l-^,^)7^1 ~ ( l - < C ^ > ) n ?
for ^rj CM*.
Proof. — Using the notations above, a calculation involving the Weyl
character formula implies that
-r r\ ri+2j_l/n-2+j\ . . . , . . ,^ A U ) = ——^~j— ( • J ? Ior an positive integers ^.
72 1 \ ^ /
918 K. OELJEKLAUS, P. PFLUG & E.H. YOUSSFI
See ([FH], pp. 267-315). In addition, some computing shows that a = 2n—2.
See ([FH], pp. 399-414). The lemma now follows from (2.3). D
LEMMA 2.2. — The n-form on (C \ {O})714-1
n+ f—iv'-1 ^
a{z) '.= y ^ ———d^ / \ . . . /\ dzj A • • • A d^yi+i,
j=i zj
induces by restriction an SO (n + 1,C)-invariant and holomorphic n-form
a on M*.
Proof. — Let A e S0(n + 1, C), z C M* and set w = A/z. Denote by
Ajk the n x n matrix obtained from A by deleting the jth row and the A;th
column. Since A C S0(n + 1,C), Cramer's rule gives that
(2.4) a,k = (-l)fc+JdetA^.
Note also that for z € M* and Zj 7^ 0, then
(2.5) dz^-Y^^dzi onT^M*,
^j ^
where T^M* denotes the tangent space of M* at the point z. Denote by
A* a the pull-back of a. Then
71+1 f—iV-i .̂̂
(A*o;)(^) = ̂ ^ —————dw\ A • • • A dw -̂ A • • • A dwn-\-i
j=l 3
n+i /_^\j-i yi+i _^^
= 5^ ———— S detA^-fcd^i A • • • A d^ • • • A ck^+i
j=i ^ î
n+l n+1 /_^\A;+j ^^
= ^(-l)^"1 ̂  v—7——detA^cbi A • • • A dzk A • • • A c^n+i
A;=I j=i wj
THE BERGMAN KERNEL OF THE MINIMAL BALL 919
by (2.4) = ̂ (-l^-1 ̂  ̂ dz, A . . . A dz, A ... A dzn^
k=l j=l w^
n+1 n+1
by (2.5) ^^^l^^^^A.-.A^-iA^AfV^)
^=1 j=i ̂  v^^' 7
A ck^+i A • • • dzk A • • • A dzn-}-i
n+1 n+1
= E^1)'"1 J^W^Zkdz, A ... A dT, A ... A d^+i
fc=l j=l ^3
n+1 //_^j-l n+1 \ ^
= E „, ^ E a^^ ^l A • • . A d^- A • • • A d^+i
,=1 \ W3Z3 k=l )
71+1 . 1 ^
= E^"1)'7"1—^! A • • • A dzj A • • • A dzn+i = a{z).
j=i ^
That the restriction of a to M* is holomorphic can be seen by evaluating
the form a on the n-fold exterior power of the tangent space. D
3. Proper holomorphic mappings from M into C71.
Consider the projection pr : C7^1 —^ C71 defined by
Pr(zi,'",Zn+l) := (z^'-'.Zn).
The restriction F := pr\M ofpr to M gives a proper holomorphic mapping
of degree 2 from M onto B^. Let W be the branching locus of F and V
the image of W under F. Denote by y? and '0 the two local inverses of F
defined for z = (^i, • • • , Zn) C B^ \ V by
(̂ ) = (^^•2Q
'0(^) = (Z,-Z^»^).
LEMMA 3.1. — Ify? := (^i, .- . ,^+i) and^ := (^i , - - - ,^+i) are
the local inverses ofF defined on B^ \ V, then
(3.1) ^*(a) = -^—^(-l)^^ A . . . A dzn
1'^ Z ® Z
(3.2) ^*(a)=__tZL(_i)"^/\.../\^
920 K. OELJEKLAUS, P. PFLUG & E.H. YOUSSFI
Proof. — The pull back of a under (p is
n+l^(-Ip-i __
^ (a) = ̂  ———•—dw^ A • • • A dwj A • • • A dw^i(a) 
. ^3j=i J
V- (-1P~1 — (-I)71= JL —„,——dz^^ • • • Ad^ Ad^ Adw„+l+-v——^-cbiA • • • /\dzn.
j=l w? ^n+l
But for 1 < j < n
(-lp-1
dz\ A • • • A d -̂ A • • • A d^ A d(pn+i
(-I?-1 — / n
J——cbi A . • • A dzj A • • • A dzn A f - V" -^-d^)dz-i • ( ^_
fc=
dzi A • • • A dzj A • • • A dzn A d^-
'̂ '< ———/. Wr,-J-1 /
(-IP-
'J v ^wn+l
^n+1
(_l)J+n-J
d2:i A • • • A dzj A • • • A cb^
Wn+l
dzi A • • • A dz,=(-i)
Wn+l
Thus
^*(a) = fL2^ + (-1)"-"- \ ̂  A • . • A d^
\,w»+i w»+iy
=:^(-1?A•••A^
'^^"^^•••^^
Similarily one has that
^*(a) = ———^(-l)-^i A ... A dzn.
D
If PM* denotes the Bergman projection of M* with respect to the
volume form a(z)/\a(z), and ifP®, denotes the Bergman projection ofB^,
then we have the following transformation rule
LEMMA 3.2. — Ify := (^ i , - ' - ,^+i) and^ := (^i, • • • ,^n+i) are
the JocaJ inverses ofF denned locally on B^ \ V, then
PM- {zn^{h o F))(z) = Zn^((P^h) o F)(z)
for all h e ^(B^), where V is the image of the branching locus ofF.
THE BERGMAN KERNEL OF THE MINIMAL BALL 921
Proof. — First observe that the lemma holds for all holomorphic
functions h G L2^). Indeed, by virtue of Lemma 3.1 we have that
/ \Zn^(hoF)(z)\^la{z)^^z)
JM*
= I \{zn^{hoF)){z)\2a{z)^^z)
JM*\W
= / bn+i(^(w)|V(a)(w) A y?*(a)(w)
J^\v
+ ( |^n+i(w)/i(w)|V(a)H A ^*(a)(w)
J^\v
= 2(n + I)2 { \h(w)\2 dv(w) < +00.
JBAV^\v
Thus Zn^(hoF)(z) € L^M*,^) Aa(^)).
Next let / e L2(M*,a(^) A Q/(2;)) be a holomorphic function, and let
g be an element of the space C§°(^ \ V) of all C'°°-function with compact
support in B^ \ V. Then
L^^^^y^^
=(n^\f ̂ J^J^^^f (^^^)^(J
IA. ^n+l(w) ^W/ ^ ^n+l(w) 9w/ v 'J
so that by integration by parts we obtain that
M^1^0^)
Since the space
PM (^+i(—— o F)\ = 0, for all j = 1, • • • ,n.
/T_/ ._ J °J . ,, .̂ . /^CX3/-n3) \ t7-\ \.
u'~\~9w,'9^CQ ̂ \v^
is dense in the orthogonal complement in L^B,,) of the subspace L^(B^)
of all square integrable holomorphic functions on B^, the lemma follows. D
LEMMA 3.3. — If(p and ^ are as before, then
z^K^ (F(.), w) = (n + I)2 '^^^)) + W^W)} ,
^n+l(w) ^n+l(w) J
2^ G M*,w C B,,.
Proof. — Let w C B^ \ V and let r > 0 be chosen so small that
w + rA" C B^ \ V, where A is the unit disc in C. By Remark 6.1.4 in [JP],
922 K. OELJEKLAUS, P. PFLUG & E.H. YOUSSFI
there is a C°°-function u : C7'1 -^ [0, +00) with compact support in w+rA1'1
such that
P^u = K^(',w).
By virtue of Lemma 3.2 we see that for z € M*,
Zn^K^(F{z),w)=Zn^{P^u)(F(z))=PM^Zn^l(uoF){z))
= ( C^i(noF)(C)^M-(^C)^(C)Ao(0
JM^
=(n+l)2 / uW "^£^))+^^^))" ̂
^B. _ ^n+lW ^n+l(77)
_ . 2/^M*(^^(W)) KM^(Z^(W))\
— { T l - f l ) I — — . r 1 -———7—r- ) ?
^ ^n+l(^) ^n+l(w) /
and the lemma is proved. D
4. Proof of the main result.
Proof of the theorem. — For z, w G B^ \ V, set
f s := 1- < ^,w >, t := ^n+i0)^n+i(w)
[ a; :=< 2:, w > -\-t and 2/ :=< z^w > —t.
Then using the notations in the main theorem we have X == s and Y = t2.
By Lemma 2.1 we see that for some positive constant C we have
K- ̂ - ̂  ̂ ((T^T-^)
^.(^),^))=c((^,-^),
so that by Lemma 3.3 we obtain that
K^ (z,w)= 4C(n + I)2 f(x) ~ f(y), wherex y
f( \= 2n n+lJw (l-u)"+1 (1-u)"'
On the other hand,
f(x) - f(y) (s +1)"+1 - (a - ̂ )"+1 n+l(s+t)n-(s-1)"
X-y -n ((s2-^2)n+l 2 ^S2-^)"
THE BERGMAN KERNEL OF THE MINIMAL BALL 923
and
(a +1)"-+1 - (s - f)"+1 _ (s + ̂ )"+1 - s"+1 (s - ̂ )»+i - g»+i
* ~ t + ^
=\-(n+l\."+1-^-1
^v J /
n+1 , -,
+E(^ )^+1-J(-^-1
n+l
=E(r^•)s"+l-J[^ l+(-^- l]j=i J
=E(^i)sra-fc[tfc+(-^]re^u
[¥1
-aV^^+^s^afc^fc
-^^k+l)8 t •
fc=0
Similarily we have that
(,+^-(g_^y „ \n-l-2^
t -2 z. V2fc+ir * •
fc==0
Therefore,
2ny[?] ( n^l}^-2k.2k
f{x)-f(y) ^^^-Q^fc+ir t
x - y (s2 - ̂ 2)7^+l
__ r n—l i / /yi \
, .^ '̂(2^l)••-'-'t''t
-('•+1)—————(.TTTi)^—————•
n , / " \ n-2k ( n + l \ _,But(2fc+l)=-^-^(2^+l)•Thus
2nV[!!] ^n+ l^c"-2^2fc/ (^)- / (y)_ ^^^fc+ir r
a; - y (s2- f2)^1
ES'̂  - 2fc) Q^ lJ.ra-l-2^(^ - f2)
- ( s 2 - ̂ )"+1
_ Ell'o (2^ ̂  i)."-1-2^2'1^ - (n - 2A-)(.2 - t2)]
- (s2 - t 2 ) » + 1 •
924 K. OELJEKLAUS, P. PFLUG & E.H. YOUSSFI
It follows that
K^ (z, w)
E^" (2^ t i )xn-l-2^[2^ - (n - WX2 - Y)}
= 4C(n + I)2————^+-^_______________________
(JP-y)7^1
where X and Y are as in the statement of the theorem. To compute the
constant we use the formula
1= / K^(0,w)dV(w).
JB.
D
5. Applications.
THEOREM 5.1. — Let D C C71 be a pseudoconvex domain with
C2f-boundary and let f : D —^ B,, be a proper holomorphic mapping. Then
Af C V{f) where
A = Af := [z e D : f{z) • f(z) = 0}, V(f) := [z € D : det f\z) = 0}.
Proof. — Observe that A is an analytic subset of D. Assume that
there exists a point a € A with det/'(a) ^ 0. To get a contradiction it
suffices to show that:
if 2^ € A, ̂  -^ z° e 9D, then det /'(^) -^ 0.
We choose a ball B{z°,s) , 0 < rj < 1 so that r](n + 2) > n + 1,
and a defining function r of D D B{z°,s) such that F :== -(-r)77 is
plurisubharmonic on D H 5 (2:°, 5); this can be achieved using a result of
Diederich-Fornaess [DF]. Moreover, we may assume that /(^) —^ w° e
E n 9B^, where B := {<• e C" : C • C = 0}.
Assuming that / is a mapping with multiplicity m, we know by
Pinchuk [Pi2] that
mK^z) ̂  \detff(z)\2K^(f{z)J(z))^ z ^ D.
It is well known that Ko(z,z) < ddist (^D)-^1), z C D. Hence we
get
|det/'^)|2 < G2(^(/(^J(^))-ldist(^^)-(n+l).
THE BERGMAN KERNEL OF THE MINIMAL BALL 925
Now we apply the theorem to obtain that
Idet/'^)]2 ^ €3(1 - l/^^l'r+'/dist^^D)^1, v » 1.
Fix s ' < s and define on D the following function:
. . ._ f max{r^), 1/z - z°\2 - s'2} if z e D U B(^),
w ' ll^-^l2-^2, if^eDYB^0^').
It is clear that v is plurisubharmonic on D and that i?(^) = ?(z) for
2; G £) H B(^°, 5"), 0 < s" < s ' sufficiently small.
For w € B* we put p(w) :== max{z»(^) : ^ € D^f(z) = w}.
Obviously, p is plurisubharmonic on B*. In particular, for v » 1 we have
p{f{zy))>v(zy)=^).
Exploiting that ®* is balanced and the Hopf-Lemma on H D B^ leads
to the following estimate: p{f{z^) < C^f^z^2 - 1), v » 1; C^ > 0
independent of z^. Therefore
|detn^)|2 <G5(-r(^))^n+2)/dist(^,9D)n+l -. 0, if 1^ ̂  oo,
which leads to that contradiction we mentioned at the beginning of the
proof. D
COROLLARY 5.2. — There are no unbranched proper holomorphic
mappings from D onto B^ for any bounded pseudoconvex domain with a
C2- boundary; in particular, such a D is never biholomorphically equivalent
to®*.
Moreover, if D is assumed to be strongly pseudoconvex we get even
more:
THEOREM 5.3. — Let D C C71 be a strongly pseudoconvex domain
with C2- boundary. If f : D —^ B* is a proper holomorphic mapping, then
A = V{f).
Proof. — Assume the inclusion V(f) C A is not correct. Then, by
the maximum principle, there is a sequence z^ € V(/), z^ —^ z° e QD
such that \f(z^) • / ( z ^ ) } > C > 0. Without loss of generality we assume
that f{z^) —> w°. Since |w° • w°| > 0 we conclude that w° is a strongly
pseudoconvex boundary point of B*. By Theorem 3 of [Ber] there is a
neighborhood U = U(z°) such that / extends to a continuous mapping on
U H D. Then using Theorem 3' of [Pil] we obtain that / € ^{D H U).
Finally using Theorem 1 of [Pi2] we finally get the contradiction to the fact
that z° C 7(7) • Q
926 K. OELJEKLAUS, P. PFLUG & E.H. YOUSSFI
We recall that a bounded domain ^ is said to satisfy condition (Q)
if the Bergman projection of Q maps C§°(fl.) into the space 0(H) of all
holomorphic functions on a neighborhood of F2. It was proved recently in
[Th] that Q satisfies condition Q is if and only if that for every compact
subset L of f2, there is an open neighborhood U = U(L) of Q such that
the Bergman kernel K^{z,w) of fl extends to be holomorphic on U as a
function of z for each w € L, and K^ is continuous on U x L.
LEMMA 5.4. — The ball B^ satisfies condition (Q).
Proof. — For z^ w e B^, we have
(1- < ;Z,W >)2 - (Z9 z)(ww) ^|1- < ^,W > |2 - ̂ •2;||ww|
>(1 — |^| |w|)2 — \Z • Z\\W •w|
>fl- |^ | |w|- (l — \z\\ \ — ^/\z • z\^/\w •w|)
> (l - ̂ /\z\2 + \z • z\^/\w\2 + \w • w\\
where the last inequality holds because of Cauchy-Schwarz's inequality.
Therefore for some positive constant C we have
C
\ K ^ ( z , w ) \ < — — — ^ ^ , for all z, w e B^.
[1 - ̂ /\z\2 + |<^•2;|v / |w|2+ \ww\\
This shows that ®^ satisfies condition (Q). D
THEOREM 5.5. — Let D C C71 be an arbitrary bounded circular
domain which contains the origin.
(1) If f : B^ —^ D is a proper holomorphic mapping, then f extends
holomorphically to a neighborhood ofB^.
(2) IfD is smooth then there is no proper holomorphic mapping from ]B»<
into D.
Proof. — Since, by Lemma 5.4, B^ satisfies condition Q, part (1) of
the theorem becomes a consequence of Theorem 2 of [Bel]. To see that
part (2) of theorem holds, it is enough to notice that if there is proper
holomorphic mapping / : B^ —^ D, and if Q is a defining function of D,
then Q o f is a defining function for B^, which will imply that B^ is smooth
and thus leads to a contradiction. D
THEOREM 5.6. — Let L be a compact subset of B^ and let C
be a boundary point of B,,. Then every holomorphic function f in a
THE BERGMAN KERNEL OF THE MINIMAL BALL 927
neighborhood ofL is the uniform limit of functions in the complex span of
the functions
^^(.,C), /^N?.
Proof. — Since B^ is a Runge domain and satisfies condition (Q) the
proposition follows from Theorem 2.5 of [Th]. D
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holomorphes propres, Banach Center Publications, 31 (1995), 91-98.
[DF] K. DIEDERICH and J.E. FORNAESS, Pseudoconvex Domains: Bounded Strictly
Plurisubharmonic Exhaustion Functions, Inventiones Math., 39 (1977), 129-
141.
[FH] W. FULTON & J. HARRIS, Representation theory, Graduate Texts in Math.,
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[HP] K.T. HAHN & P. PFLUG, On a minimal complex norm that extends the euclidean
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[JP] M. JARNICKI & P. PFLUG, Invariant Distances and Metrics in Complex Analysis,
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[Ki] K.T. KIM, Automorphism groups of certain domains in C^ with singular bound-
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[Le] J.J. LOEB, Les noyaux de Bergman et Szego pour des domaines strictement
pseudo-convexes generalisent la boule, Publicationes Math., 36 (1992), 65-72.
[OY] K. OELJEKLAUS & E.H. YOUSSFI, Proper holomorphic mappings and related
automorphism groups, J. Geom. Anal., to appear.
[Pil] S. PINCHUK, Scaling method and holomorphic mappings, Proc. Symposia in Pure
Math., Part 1, 52 (1991).
()2^ K. OELJEKLAUS, P. PFLUG & E.H. YOUSSFI
[Pi2] S. PINCHUK, On proper holomorphic mappings on strictly pseudoconvex domains
Sib. Math. J., 15 (1974).
[Th] A. THOMAS, Uniform extendability of the Bergman kernel, Illinois J Math 39
(1995), 598-605. ' ''
Manuscrit recu Ie 18 septembre 1996,
accepte Ie 20 decembre 1996.
K. OELJKLAUS & E.H. YOUSSFI,
Universite de Provence
Centre de Mathematiques et d'lnformatique
39 rue F. Joliot-Curie
13453 Marseille Cedex 13 (France).
karloelj@gyptis.univ-mrs.fr
youssfi@gyptis.univ-mrs.fr
P. PFLUG,
Universitat Oldenburg
Fachbereich Mathematik
Postfach 2503
26111 Oldenburg (Allemagne).
PFLUG@mathematik.uni-oldenburg.de


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