2. Hartogs Pseudoconvexity

1. Hartogs Extension Theorem

The following is a classical result known as Hartogs Extension Theorem.

Theorem I. ([2]) Let n≧ 2 and suppose that 0 < rj< 1 for 1≦j≦n. Then every holomorphic function f on the domain

H(r) = {z∈Cⁿ; |zj| < 1 for 1≦j≦n−1, rn< |zn| < 1}

∪{z∈Cⁿ; |zj| < rj for 1≦j≦n−1, |zn| < 1}.

has a unique holomorphic extension ˜f to the polydiscΔ(0 ; 1).

This theorem leads us to the following which has somewhat excessive assumptions, and which induces, in the next section, more general notions of pseudoconvexity than Hartogs one.

Theorem I′. Let n≧2 and 0 < ri< 1 for 1≦i≦n, and we set

H(r) = {z∈Cⁿ; |zi| < rifor 1≦i≦q, |zj| < 1 for q + 1≦j≦n}

Hk={z∈Cⁿ; |zi| < 1 for 1≦i≦q, |zj| < 1 for q + 1≦j≦n and j ≠k, rk< |zk| < 1}, for q + 1≦k ≦n.

Then every holomorphic function f on the domain H = H(r) ∪

( ∪

⁾

2. Hartogs Pseudoconvexity

The following defnitions are seen in [2]. We call the pair (Γ,＾Γ) of the following compact sets Γ,＾Γ ⊂

（

）

Akita University（Natural Science）

62，1−8（2007）

Remarks on Hartogs Pseudoconvexity of Domains in C

Toshio UDA

Abstract

We remark Hartogs pseudoconvexity of domains in Cⁿ. The notion of Hartogs q-pseudoconvexity is introduced, in this paper, and we study the relationship between various pseudoconvexity.

n k=q+1

Cⁿthe Hartogs frame in Cⁿ.

Γ0 = {z∈Cⁿ; zi= 0 for 1≦i≦n−1, |zn|≦1},

Γn = {z∈Cⁿ; zi= 0 for 1≦i≦n−2, |z_n−1|≦1, |zn| = 1}, Γ = Γ0∪ Γn,

＾Γ = {z∈Cⁿ; zi= 0 for 1≦i≦n−2, |zj|≦1 for n−1≦j≦n}.

A pair (Γ*, ＾Γ*) of compact sets Γ*,＾Γ* ⊂ Cⁿis called a Hartogs figure if there exists a biholomorphic map F :＾Γ→＾Γ*, such that F(Γ) =Γ*.

The following result is obvious by Theorem I.

Proposition 1. ([2]) Let (Γ*, ＾Γ*) be a Hartogs figure. Then every f∈O(Γ*) has a holomorphic extension

˜f ∈O(＾Γ*).

A domain D⊂ Cⁿis called Hartogs pseudoconvex if for every Hartogs figure (Γ*, ＾Γ*) with Γ* ⊂ D one has ＾Γ* ⊂D as well.

Now, according to Theorem I′, we shall define more general notions. Let 0 < q≦n−1. The pair (Γ,＾Γ) of compact sets Γ,＾Γ⊂ Cⁿdefined as follows is called the Hartogs q - frame in Cⁿ.

Γ0 = {z∈Cⁿ; |zi| = 0 for 1≦i≦q, |zj| ≦1 for q + 1 ≦j ≦ n},

Γk = {z∈Cⁿ; |zi| = 0 for 1≦i≦q−1, |zj| ≦1 for q ≦j ≦ n and j≠k, |zk|＝1 }, for q＋1≦ k≦n

Γ = Γ⁰∪

( ∪

⁾

＾Γ = {z∈Cⁿ; |zi| = 0 for 1≦i≦q−1, |zj|≦1 for q≦j≦n}.

A pair (Γ*, ＾Γ*) of compact sets Γ*, ＾Γ* ⊂Cⁿis called a Hartogs q - figure if there exists a biholomor- phic map F : ＾Γ→ ＾Γ*, such that F(Γ) = Γ*.

A domain D⊂Cⁿis called Hartogs q - pseudoconvex if for every Hartogs q - figure (Γ*, ＾Γ*) with Γ*

⊂D one has ＾Γ* ⊂D as well. The case q = n−1, it is Hartogs pseudoconvex.

By Theorem I′we have

Proposition 2. Let (Γ*, ＾Γ*) be a Hartogs q - figure. Then every f∈O(Γ*) has a holomorphic extension

˜f ∈O(＾Γ*).

Hartogs q - pseudoconvexity is a notion which is weaker than Hartogs pseudoconvexity as follows.

Proposition 3. Every Hartogs pseudoconvex domain is Hartogs q - pseudoconvex for 0 < q≦n−1.

n k=q+1

Proof. Let D⊂Cⁿbe a Hartogs pseudoconvex domain, (Γ′, ＾Γ′) the Hartogs frame and (Γ,＾Γ) the Har- togs q - frame. For a biholomorphic map F :＾Γ → F^〜 (＾Γ) with F(Γ) ⊂D, we shall show that F(＾Γ) ⊂D.

For any fixed (ζ⁰q, . . . ,ζ⁰n−2), |ζ⁰j|≦1, it is suffcient to show that if |ζn−1|≦1 and |ζn|≦1, then we have

F(0″,ζ⁰n−1,ζ⁰q＋1, . . . , ζ⁰n−2,ζ⁰q,ζn) ∈D, where 0″= (0, . . . , 0) ∈C^q−1.

Let G be a biholomorphic map defined by

G : (0″, 0, . . . 0, ζn−1,ζn) (0″,ζn−1,ζ⁰q＋1, . . . ,ζ⁰n−2,ζ⁰q,ζn).

We set ζn−1= 0 and |ζn|≦1, then G(Γ′⁰) ⊂ Γ⁰. We set |ζn−1|≦1 and |ζn| = 1, then G(Γ′ⁿ) ⊂ Γⁿ. So we have G(Γ′) = G(Γ′⁰∪Γ′ⁿ) ⊂ Γ⁰∪ Γⁿ⊂ Γ. It follows, for a biholomorphic map F G, F G(Γ′) ⊂ F(Γ) ⊂ D. Since D is Hartogs pseudoconvex, we have F G(＾Γ′) ⊂ D. Hence, for |ζn−1|≦ 1 and

|ζn|≦1, we have

F(0″,ζn−1,ζ⁰q＋1, . . . , ζ⁰n−2,ζ⁰q,ζn) = F。G(0″, 0, . . . , 0, ζn−1,ζn) ∈F。G(＾Γ′) ⊂D

Thus D is Hartogs q - pseudoconvex.

More generally, we have

Proposition 4. Let q′< q. If D⊂Cⁿis Hartogs q - pseudoconvex, then D is Hartogs q′- pseudoconvex.

Proof. Let (Γ′, ＾Γ′) with Γ′= Γ′⁰ ∪(

∪

F(0″,ζq,ζ⁰q′＋1, . . . ,ζ⁰q−1,ζ⁰q′,ζq＋1, . . . ,ζn) ∈D,

Let G be a biholomorphic map defined by

G : (0″, 0, . . . , 0, ζq, . . . ,ζn) (0″,ζq,ζ⁰q′＋1, . . . ,ζ⁰q−1,ζ⁰q′,ζ⁰q＋1, . . . ,ζn),

where 0″= (0, . . . , 0)∈C^q′−1. We set |ζj|≦1 for q≦j≦n and j≠k, and |ζk| = 1, then G(Γ^k) ⊂ Γ′^k for q + 1≦k≦n. And we set ζq= 0 and |ζj|≦1 for q + 1≦j≦n, then G(Γ⁰) ⊂ Γ′⁰. Since

G(Γ) = G

(

( ∪

⁾⁾

⁽ ∪

⁾

ｑ

︶ n−1

︶

q′

︶

q−1

︶

ｑ

︶

n k=q+1

n k=q+1 ｑ

︶

ｑ

︶

n−1︶

q′

︶

q′

︶

ｑ

︶ n

k=q_′+1

F G(Γ) ⊂ F(Γ′) ⊂ D. Hence, since D is Hartogs q - convex, we have F G(＾Γ) ⊂ D. Thus, if |ζj|≦1 for q ≦j ≦n, we have

F(0″, (0″,ζq,ζ⁰q′＋1, . . . ,ζ⁰q−1,ζ⁰q′,ζq＋1, . . . ,ζn) = F G(0″, 0, . . . , 0, ζq, . . . ,ζn) ∈F G(＾Γ) ⊂D, as desired.

A domain D⊂Cⁿis said to satisfy Theorem of continuity (C) of order q if we set, for 0 < r′< r and 0 <

ρ′< ρ,

Δ0= {z∈Cⁿ; |zi| < r′for 1≦i≦q, |zj| < ρfor q + 1≦j≦n},

Δk= {z∈Cⁿ; |zi| < r for 1≦i≦q, |zj| < ρfor q + 1≦j≦n and j≠k, ρ′< |zk| < ρ}, for q + 1≦k≦n.

Δ = {z∈Cⁿ; |zi| < r for 1≦i≦q, |zj| < ρfor q + 1≦j≦n},

then Δ⁰∪

( ∪

⁾

Proposition 5. If D ⊂ Cⁿ satisfies Theorem of continuity (C) of order q, then D is Hartogs q - pseudo- convex.

Proof. Let (Γ,＾Γ) be the Hartogs q - frame and F :＾Γ→ F^〜 (＾Γ) a biholomorphic map with F(Γ) ⊂D. For sufficiently small 0 < r′< r and for sutable ρ′< 1 < ρ, if we set

Δ0= {z∈Cⁿ; |zi| < r′for 1≦i≦q, |zj| < ρfor q + 1≦j≦n},

Δk= {z∈Cⁿ; |zi| < r for 1≦i≦q, |zj| < ρfor q + 1≦j≦n and j≠k, ρ′< |zk| < ρ}, for q + 1≦k≦n.

then we have F(Δ0) ⊂D and F(Δk) ⊂D, q + 1≦k≦n. Since D satisfies Theorem of continuity (C) of order q, a polydisc

Δ = {z∈Cⁿ; |zi| < r for 1≦i≦q, |zj| < ρfor q + 1≦j≦n}

satisfies that F(Δ) ⊂D. Hence

＾Γ= {z∈Cⁿ; |zi| = 0 for 1 ≦i≦q−1, |zj| ≦ 1 for q≦j≦n}

satisfies F(＾Γ) ⊂D. Thus D is Hartogs q - pseudoconvex, as desired.

Now, D⊂Cⁿis said to have C^kboundary at p ∈∂D if there exist an open neighborhood U of p and a

n k=q+1

ｑ

︶ q′

︶ q−1

︶ q′

︶

real valued function r ∈C^k(U) with the following properties:

U∩D = {z ∈U ; r(z) < 0} and dr(z)≠0 for z ∈U.

If D has C^kboundary at every point p ∈∂D, D is said to have C^kboundary.

Notice that this includes

U∩∂D = {z ∈U ; r(z) = 0}, U−−D = {z ∈U ; r(z) > 0}.

The function r ∈C^k(U) is called a (local) defining function. If U = U(∂D), r is called a defining function.

Theorem 6. Let D ⊂Cⁿ be Hartogs q - pseudoconvex. Suppose D has C¹ boundary at p ∈∂D. Then there exists no complex n−q dimensional submanifold M such that p ∈M and M - {p} ⊂D.

Proof. We may assume n≧2. We shall show that the existence of M with the properties stated in the theorem implies that D is not Hartogs q - pseudoconvex at p ∈∂D.

Given such an M, after applying a holomorphic coordinate change in a neighborhood U of p, we may as- sume that

p = 0, M∩U = {z ∈U ; zi= 0 for 1≦i≦q, |zj| < 2δfor q + 1≦j≦n},

and that defining function r(z) for D satisfies

r(z1, . . . , zn) = xq−ϕ(z₁, . . . , zq−1, yq, zq+ 1, . . . , zn), dϕ₀= 0.

Since M∩U−{0} ⊂D, for q + 1≦k≦n,

r(0, . . . , 0, zq + 1, . . . , zn) = −ϕ(0, . . . , 0, z_{q + 1}, . . . , zn) < 0,

for |zj| < 2δ(j≠k) and 0 < |zk| < 2δ. Hence

ϕ(0, . . . , 0, z_{q + 1}, . . . , zn) ≧0, for |zj| < 2δ, q + 1≦j≦n.

The continuity of r follows that, for a sufficiently smallη> 0,

r(0, . . . , 0, zq, zq + 1, . . . , zn) < 0, (1)

where |zq+η|≦η, |zj|≦δfor q + 1≦j≦n and j≠k, |zk| =δ,

ｑ

︶

and that

r(0, . . . , 0, −η, zq + 1, . . . , zn) < 0, for |zj|≦δ(q + 1≦j≦n). (2) From (1), if we set

Kk= {(0, . . . , 0, zq, zq + 1, . . . , zn) ;

|zq+η|≦η, |zj|≦δfor q + 1≦j≦n and j≠k, |zk| =δ},

then Kk⊂D∩U for q + 1≦k≦n. And from (2), if we set

K0= {(0, . . . ,−η, zq + 1, . . . , zn) ; |zj|≦δfor q + 1≦j≦n},

then K0⊂D∩U. We set

Γ* = K0∪

( ∪

⁾

＾Γ* = {(0, . . . , 0, zq, zq + 1, . . . , zn) ; |zq+η|≦η, |zj|≦δfor q + 1≦j≦n}.

Let F : (t1, . . . , t_n) (z₁, . . . , z_n) be a biholomorphic map defined by

zi= tifor i≦q−1, zq=ηtq−ηand zj=δtjfor q + 1≦ j≦n.

Then F(＾Γ) =＾Γ* and F(Γ) = Γ*, so (Γ*,＾Γ*) is a Hartogs q - figure in U. Then we have Γ*⊂ D, while

＾Γ* ⊂/ D since 0∈＾Γ*. Thus D is not Hartogs q - pseudoconvex, as desired.

3. Levi pseudoconvexity

Let r be a defining function for D at p ∈∂D. A complex tangent space Tp^C(∂D) at p ∈∂D is given by

Tp^C(∂D) = {z ∈Cⁿ; ∂rp(z) : =

Σ

Notice that this space does not depend on the choice of the defining function r.

A domain D ⊂Cⁿ with C² boundary is called Levi pseudoconvex at p if the Levi condition holds for some C²defining function r for D near p∈∂D;

Lq(r ; z) : =

Σ

D is said to be Levi pseudoconvex if the Levi condition holds at all points p∈∂D.

n j,k=1

∂²r

∂zj∂-zk

n i=1

∂r

∂zi ｑ

︶

n k=q+1

ｑ

︶

Theorem 7. ([2]) If D ⊂Cⁿ is Hartogs pseudoconvex and D has C² boundary, then D is Levi pseudocon- vex.

A domain D⊂ Cⁿwith C²boundary is said to be Levi q - pseudoconvex at p∈∂D if there exists a sub- space E⊂Tp^C(∂D) with dim E≧n−q + 1, such that Levi condition holds on E; Lp(r ; z) ≧0 for all z∈E.

Theorem 8. Let D have C¹boundary at p ∈∂D. If there exists a complex submanifold M with dimM = n

−q in a neighborhood U of p such that p ∈M and M ∩U−{p}⊂D, then D is not Levi q - pseudocon- vex at p∈∂D.

Proof. Given a subspace E ⊂Tp^C(∂D) with dim E≧n−q + 1 . We shall show that Lp(r ; t) < 0 for some t∈E.

After applying a holomorphic change of coordinates we can assume that

p = 0, M = {z∈U ; z1= · · · = zq= 0}, E= {t∈Tp^C(∂D) ; t1= · · · = tq−1= 0}.

From the Taylor expansion of a defining function r at 0 we obtain

r(0, . . . , 0, zq + 1, . . . , zn) = 2Re h(z) + L0(r ; z) + o(|z|²) < 0,

where h(z) =

Σ

Σ

t = (0, . . . , 0, tq, . . . , tn)∈E with tq= 0 and h(t) = 0, we obtain

r(0, . . . , 0, tq+1, . . . , tn) = L0(r ; t) + o(|t|²) < 0.

Since |L0(r ; t)| = O(|t|²), if, for a sufficiently small ε> 0 and t ∈B(0 ; ε) ⊂U, we obtain L0(r ; t) < 0. Thus D is not Levi q - pseudoconvex at 0∈∂D, as desired.

References

[1] O. Fujita, Domaines pseudoconvexes d fordre g′eneral et fonctions pseudoconvexes d fordre g′ en′′eral, J.

Math. Kyoto Univ. 30-4 (1990), 637-649.

[2] R. M. Range, Holomorphic Functios and Integral Representations in Several Complex Variables, Springer-Verlag, New York (1986).

[3] M. Tadokoro, Sur les ensembles pseudoconcaves g′eneraux, J. Math. Soc. Japan, 17 (1965), 281-290.′ [4] V. V＾aj＾aitu, Some convexity properties of morphisms of complex spaces, Math. Z. 217 (1994), 215-245.

[5] V. V＾aj＾aitu, q - completeness and q - concavity of the union of open subspaces, Math. Z. 221 (1996),

n j＞q

∂r

∂zj

1 2!

n j,k＞q

∂²r

∂zj∂zk

217-229.

[6] V. V＾aj＾aitu, The analyticity of q - concave sets of locally finite Hausdorff (2n−2q) - measure, Ann.

Inst. Fourier, Grenoble, 50, 4 (2000), 1191-1203.

DEPARTMENT OFMATHEMATICS

FACULTY OFEDUCATION ANDHUMANSTUDIES

AKITAUNIVERSITY

AKITA010-8502, JAPAN