ERRATUM TO: ON L -ESTIMATES FOR ¯ COMPLETE K ¨

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ERRATUM TO: ON L2 -ESTIMATES FOR ∂¯ ON A PSEUDOCONVEX DOMAIN IN A
COMPLETE KÄHLER MANIFOLD WITH POSITIVE HOLOMORPHIC
BISECTIONAL CURVATURE
SÉVERINE BIARD
Abstract. We correct some results in our earlier paper, about the Diederich-Fornaess exponent for
the distance function in a complete Kähler manifold with positive holomorphic bisectional curvature. The
main concern is about the definition of the function τΩ . This was corrected in the author’s thesis. The
main results on the existence of Diederich-Fornaess exponent and the L2 -estimates for ∂¯ operator still hold.
We use the notations of [1].
Definition 4.2 [1] needs to be remplaced by the following:
Definition 4.2. Let U be a neighborhood of ∂Ω. Then
τΩ (x) = kΛx k2T ∗ U,hf ,
∀x ∈ U ∩ Ω.
x
x
Proposition 4.1 [1] needs to be removed. These corrections appear because of the non-continuity of the function
τΩ on the boundary ∂Ω.
We explain this confusion in the following paragraph now. This also implies different corrections in the results of
section 5 in [1].
The tools used to prove Proposition 4.1 [1] were the Cauchy-Schwarz inequality combined with a property on
the behavior of the Levi form close to the boundary but those cannot be used in this situation for the following
reasons:
Let z ∈ U ∩ Ω. Since the eigenvalue i∂ ∂¯z (−δ∂Ω )(Ln , Ln ) may be negative (δ∂Ω is not plurisubharmonic),
the Cauchy-Schwarz inequality needs to be applied on Qz = δ∂Ω (z).i∂ ∂¯z (− log δ∂Ω ) (since − log δ∂Ω is strictly
plurisubharmonic on Ω) as follows
(1)
1
1
|i∂ ∂¯z (−δ∂Ω )(Ln , Li )| ≤ Qz (Li , Li ) 2 Qz (Ln , Ln ) 2 ,
since Qz (Ln , Li ) = i∂ ∂¯x (−δ∂Ω )(Ln , Li ).
Unfortunately, this inequality is not sufficient to obtain directly the fact that τΩ < 1 on U ∩ Ω.
1991 Mathematics Subject Classification. 32W05-32T35-32Q10.
Key words and phrases. Pseudoconvex domains, Plurisubharmonic exhaustion functions, Reach, Diederich-Fornaess exponent.
1
2
SÉVERINE BIARD
This may be explained by the positivity of the curvature:
If the curvature of the manifold is zero (for example in Cn ), then τΩ is equal to 0 on the boundary from (1) since
for p ∈ W a weakly pseudoconvex boundary point and w ∈ Np , we have
d
¯
hi∂ ∂(−δ
∂Ω (zε )), w ∧ w̄i = 0,
dε
where zε = p + ε∇δ∂Ω (p) is a point ε-equidistant to the boundary.
This implies i∂ ∂¯p (−δ∂Ω )(Ln , Li ) = 0 for every Li ∈ Ker(Lp δ∂Ω ) (see [4] or the Riccati equation (Curvature
Equations) in [II, 4.2, [3]]).
This argument doesn’t hold anymore when the curvature is positive. We cannot prove the continuity of τΩ on
∂Ω by extending it by 0 on the boundary.
Hence, Corollary 4.1 in [1], converted into a Proposition is corrected as follows :
Proposition 4.1. Let (X, ω) be a complete Kähler manifold with positive holomorphic bisectional curvature and
Ω b X be a pseudoconvex domain with C 2 boundary. Then there exists a neighborhood U of ∂Ω and a positive
constant c such that
∀z ∈ U ∩ Ω,
τΩ (z) ≤ c < 1.
Proof.
Since ∂Ω is C 2 , reach(Ωc ) ≥ a > 0 for some a > 0. So there exists a neighborhood U of ∂Ω on which δ∂Ω is
also of class C 2 and on which the function τΩ is well defined. By contradiction, suppose there exists a sequence
of points zk ∈ Ω convergent to p ∈ ∂Ω such that limk→+∞ τΩ (zk ) = 1. Then, thanks to Proposition 7.2 [1], there
η
doesn’t exist an exponent η > 0 such that −δ∂Ω
is strictly plurisubharmonic on Ω, which is a contradiction with
[2].
We also correct Theorem 5.1 [1]:
Theorem 5.1. Let Ω be a relatively compact pseudoconvex domain with C 2 boundary in a complete Kähler manifold
with positive holomorphic bisectional curvature. Let ε > 0 and Vε the ε-equidistant neighborhood of the boundary
∂Ω. Then
1
t(∂Ω) ≥ 1 − inf sup τΩ2 .
(2)
ε Vε ∩Ω
Proof of Theorem 5.1.
We prove this Theorem into two steps. First, we use Ohsawa and Sibony’s idea [2] and precise a first estimation,
thanks to Proposition 5.1 [1]. Secondly, we deduce an estimation of the order of plurisubharmonicity.
ERRATUM
3
1
Step 1: Prove that on a small enough neighborhood U of Ω, for all 0 < α < inf U ∩Ω γ.(1 − τΩ2 ), the function
α
−δ∂Ω
is strictly plurisubharmonic on U ∩ Ω.
1
Remark that we have inf U ∩Ω γ.(1 − τΩ2 ) > 0:
By Proposition 4.1, there exists a neighborhood U 0 of ∂Ω and a constant c0 > 0 such that τΩ (x) ≤ c0 < 1 for
1
√
x ∈ U 0 ∩ Ω. Hence, c0 (x) := 1 − τΩ2 ≥ 1 − c0 := c > 0 on U 0 ∩ Ω.
Moreover, the fonction γ being continuous on U 0 ∩ Ω̄ and equal to 1 on ∂Ω, reducing U 0 if need be, inf U 0 ∩Ω γ > 0.
Let (Ui )1≤i≤N be a small enough covering of ∂Ω. By Proposition 5.1 [1], for all x ∈ Ui ∩ Ω,
(3)
Qx (v) ≥ c0 (x).
n−1
X
!
2
CΩ δ∂Ω
(x)|vi |2
2
+ γ(x).|vn |
.
i=1
α
¯
By computing i∂ ∂(−δ
∂Ω ) for α > 0 and by Proposition 5.1 [1], we deduce a sufficient condition on α in order
α
for the function −δ∂Ω
to be strictly plurisubharmonic:
From (3), we have on U ∩ Ω for U =
(4)
S
1≤i≤N
Ui ∩ U 0 :


n−1
X
C
γ(x)
1
Ω
2

CΩ |vj |2 +
+ 2
|vn |2 
2 (x) Qx (v) + CΩ |vn | ≥ c0 (x)
δ∂Ω
c
(x)
δ
(x)
0
∂Ω
j=1


n
n−1
X
X
1
1
C
γ(x)
Ω
⇐⇒
aij vi v¯j + 2 k∂δ∂Ω k2 |vn |2 ≥ c0 (x) 
CΩ |vj |2 +
+ 2
|vn |2  .
δ∂Ω (x) i,j=1
δ∂Ω
c
(x)
δ
(x)
0
∂Ω
j=1
Let α > 0 and v ∈ T (1,0) U ,
α
¯
hi∂ ∂(−δ
∂Ω ), v ∧ v̄i =
=
=
=
≥
(5)
≥
α−1
α−2
¯
¯
hαδ∂Ω
i∂ ∂(−δ
∂Ω ) − α(α − 1)δ∂Ω i∂δ∂Ω ∧ ∂δ∂Ω , v ∧ v̄i
¯
(1 − α)
i∂ ∂(−δ∂Ω )
α
¯
αδ∂Ω
i∂δ∂Ω ∧ ∂δ∂Ω , v ∧ v̄ +
, v ∧ v̄
2
δ∂Ω
δ∂Ω


n
X
1
(1
−
α)
α 
αδ∂Ω
aji vj v¯i +
k∂δ∂Ω k2 |vn |2 
2
δ∂Ω j,i=1
δ∂Ω
1
α
α
2
2
2
αδ∂Ω
2 Qx (v) + CΩ |vn | − δ 2 k∂δ∂Ω k |vn |
δ∂Ω
∂Ω


n−1
2
X
C
γ(x)
α.k∂δ
k
Ω
∂Ω
α 
c0 (x)αδ∂Ω
CΩ |vj |2 +
+ 2 −
|vn |2  by (4)
2
c
(x)
δ
c
(x).δ
0
0
∂Ω
∂Ω
j=1
α
Cα (x).δ∂Ω
n
X
|vj |2 ,
i=1
where Cα is a positive constant dependent on α.
si
CΩ
γ(x) α.k∂δ∂Ω k2
+ 2 −
> 0,
2
c0 (x)
δ∂Ω
c0 (x).δ∂Ω
4
SÉVERINE BIARD
Assumption (5) is equivalent to 0 < α <
2
γ(x).c0 (x) + CΩ .δ∂Ω
(x)
.
2
k∂δ∂Ω k
As k∂δ∂Ω (x)k = 1 on U small enough, if
0<α<
2
inf (γ(x).c0 (x) + CΩ .δ∂Ω
(x)) =
x∈U ∩Ω
inf (γ(x).c0 (x))
x∈U ∩Ω
then
α
α
2
¯
hi∂ ∂(−δ
∂Ω ), v ∧ v̄i ≥ Cα δ∂Ω kvk
(6)
on U ∩ Ω.
Step 2: Estimate of the order of plurisubharmonicity t(∂Ω).
Denote Vε the ε-equidistant neighborhood of the boundary ∂Ω and U the previous neighborhood. There exists
ε0 > 0 such that for all 0 < ε < ε0 , Vε ⊂ U .
α
By Step 1, for all 0 < α < inf x∈Vε ∩Ω γ(x).c0 (x), the function −δ∂Ω
is strictly plurisubharmonic on Vε ∩ Ω.
So,
∀ 0 < ε < ε0 ,
t(∂Ω) ≥
inf
x∈Vε ∩Ω
γ(x).c0 (x) .
And,
t(∂Ω) ≥ sup
0<ε<ε0
inf
γ(x).c0 (x) .
inf
γ(x)).(1 − inf sup τΩ2 (x)).
x∈Vε ∩Ω
But,
sup
ε
inf
x∈Vε ∩Ω
γ(x).c0 (x) = (sup
ε
1
x∈Vε ∩Ω
ε x∈Vε ∩Ω
Recall that γ is continuous U ∩ Ω̄, so supε inf x∈Vε ∩Ω γ(x) = inf x∈∂Ω γ(x) = 1.
Hence,
1
t(∂Ω) ≥ 1 − inf sup τΩ2 (x).
ε x∈Vε ∩Ω
The correction of Corollary 4.1 in [1] implies the following modification of Corollary 5.1 in [1] :
Corollary 5.1. Let (X, ω) be a complete Kähler manifold with positive holomorphic bisectional curvature and
Ω b X, a pseudoconvex domain. Let (Ωk )k≥1 be an exhaustion by smooth pseudoconvex domains of Ω such that
sup τΩk ≤ c < 1,
k
for a positive constant c independent of k. Then there exists α > 0 such that for all k ≥ 1, there exists a neighα
borhood Uk of ∂Ωk such that −δΩ
∈ s − P SH(Uk ∩ Ωk ).
k
ERRATUM
5
Proof. Since for all k ≥ 1, Ωk b Ω is a pseudoconvex domain with C 2 boundary. We also have
¯ log δ∂Ω ) ≥ CΩ ω,
i∂ ∂(−
k
on Ωk .
By using the proof of Proposition 5.1 on Ωk , there exists a neighborhood Uk on which any αk with
1
0 < αk ≤ (1 − τΩ2k ) min(CΩ , γk ),
is a Diederich-Fornaess exponent for the distance function δΩk on Uk ∩ Ωk .
Since γk is continuous and equal to 1 on ∂Ωk , reducing Uk if need be, there exists a positive constant b such
that for every k ≥ 1, γk ≥ 1 − b on Uk ∩ Ωk . Moreover, by assumption, there exists a positive constant c̃ such
1
that for every k ≥ 1, 1 − τΩ2k ≥ c̃ > 0 on Uk ∩ Ωk . So, there exists an exponent α such that for every k ≥ 1, there
α
exists a neighborhood Uk of ∂Ωk such that −δ∂Ω
∈ s − P SH(Uk ∩ Ωk ).
k
Aknowledgment: The author would like to thank J-P. Demailly and M. Adachi for pointing out some errors
in the paper [1].
References
1. S. Biard, On L2 estimates for ∂¯ on a pseudoconvex domain in a complete Kähler manifold with positive holomorphic
bisectional curvature, The journal of Geometric Analysis 24 (2014), no. 3, 1583–1612.
2. T. Ohsawa and N. Sibony, Bounded P.S.H functions and pseudoconvexity in Kähler manifold, Nagoya Mathematical Journal
149 (1998), 1–8.
3. P. Petersen, Riemannian geometry, vol. GraduateTexts in Mathematics 171, Springer, 2000.
¯
4. E. Straube, Good Stein neighborhood bases and regularity of the ∂-Neumann
problem, Illinois Journal of Mathematics 45
(2001), no. 3, 865–871.
Department of Mathematics, Mailstop 3368, Texas A& M University, College Station, TX 77843-3368, USA
E-mail address: [email protected]
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