FINITE TYPE CONDITIONS ON REINHARDT DOMAINS

SIQI FU ALEXANDER V. ISAEV STEVEN G. KRANTZ

Abstract. In this paper we prove that, if p is a boundary point of a smoothly bounded pseudoconvex Reinhardt domain in ℂ^{n}, then the variety type at p is identical to the regular type.

In this paper we study the finite type conditions on pseudoconvex Reinhardt domain. We prove that, if p is a boundary point of a smoothly bounded pseudoconvex Reinhardt domain in ℂ^{n}, then the variety type at p is identical to the regular type. In a forthcoming paper, we will study the biholomorphically invariant objects (e.g., the Bergman kernel and metric, the Kobayashi and Carathéodory metrics) on a pseudoconvex Reinhardt domain of finite type.

We first recall some definitions. A domain Ω ⊂ ℂ^{n} is Reinhardt if

(e^{iθ_{1}}z_{1}, …, e^{iθ_{n}}z_{n}) ∈ Ω

whenever (z_{1}, …, z_{n}) ∈ Ω and 0 ≤ θ_{j} ≤ 2π, 1 ≤ j ≤ n. Denote

Z_{j} = {(z_{1}, …, z_{n}) ∈ ℂ^{n}; z_{j} = 0},

for j = 1, …, n. Let Z = ⋃_{j=1}^{n} Z_{j}. Define L: ℂ^{n} \ Z → ℝ^{n} by

L(z_{1}, …, z_{n}) = (log |z_{1}|, …, log |z_{n}|)

and

L^{∗}(z_{1}, …, z_{n}) = (log z_{1}, …, log z_{n}).

where, in the second case, the logarithm takes the principle branch, and L^{∗} is defined locally near every point of ℂ^{n} \ Z.

1991 Mathematics Subject Classification. Primary 32F25, Secondary 52A50.

Key words and phrases. Reinhardt domain, pseudoconvex, finite type.

Research supported in part by a grant from the National Science Foundation. Research at MSRI supported in part by NSF grant #DMS 9022140.

Let Ω = {z ∈ ℂ^{n}; ρ(z) < 0} be a bounded domain with smooth boundary. A boundary point p ∈ bΩ is of finite variety type m if

(1)  m = sup \{\frac{v(ρ ∘ f)}{v(f)}\} < ∞

where the supremum is taken over all analytics disc f: Δ → ℂ^{n} such that f(0) = p. We use v(f) to denote the order of vanishing of f at 0 (see [K] for details on this notion). When the supremum in (1) is taken over all regular analytic discs f (i.e., f′(0) ≠ 0), then m is called the regular type of p; when the supremum in (1) is taken over all complex lines through p, then m is called the line type of p.

It has been proved by McNeal [M] that, for a boundary point of a smoothly bounded convex domain, the variety type is identical to the line type. This result was generalized by Boas and Straube [B-S] to a star-like boundary point (see Theorem 4 below for the exact statement). Meanwhile, J. Yu [Y] showed that the Catlin multi-type is identical to the D’Angelo q-type for a boundary point of a convex domain. In this paper, we shall prove that, for a boundary point of a pseudoconvex Reinhardt domain, the regular type is identical to the variety type.

Lemma 1. (Reinhardt) If Ω is a pseudoconvex Reinhardt domain, then L(Ω \ Z) is convex. Furthermore, if Ω ∩ Z_{j} ≠ ∅, then (z_{1}, …, z_{j−1}, λz_{j}, z_{j+1}, z_{n}) ∈ Ω whenever (z_{1}, z_{2}, …, z_{n}) ∈ Ω and |λ| ≤ 1.

Lemma 2. Let Ω be a smoothly bounded pseudoconvex Reinhardt domain and let p ∈ bΩ. Then there is a neighborhood U of p such that bΩ∩U has a plurisubharmonic defining function ρ(z).

Proof . If p ∈ Ω \ Z, then L(z) is well-defined in a neighborhood of p. Since L(Ω \ Z) is convex, there is a local convex defining function ρ̃(u_{1}, …, u_{n}) of L(Ω \ Z) near L(p). Let ρ = ρ̃(L(z)). Then ρ is a plurisubharmonic defining function of bΩ near p.

If p ∈ Ω ∩ Z, then, without lose of generality, we may assume that

p = (p_{1}, …, p_{k}, 0, …, 0),

with p_{j} ≠ 0, 0 ≤ j ≤ k. Let σ(z) be the Euclidean distance from z to bΩ. Then σ(z) is smooth in a neighborhood of bΩ and satisfy σ(z_{1}, …, z_{n}) = σ(e^{iθ_{1}}z_{1}, …, e^{iθ_{n}}z_{n}). It follows that \frac{∂σ}{∂z_{j}} (z_{1}, …, z_{n}) = 0 whenever z_{j} = 0. Since \frac{∂σ}{∂z_{j}} (p) = 0, k + 1 ≤ j ≤ n, one of \frac{∂σ}{∂z_{j}} (p) , 1 ≤ j ≤ k, must not be zero.

For simplicity of notation, we may assume that \frac{∂σ}{∂z_{1}} (p) ≠ 0. Therefore bΩ is locally defined by an equation of the form |z_{1}| + f(|z_{2}|, …, |z_{n}|) = 0 near p. This in turn implies that L(bΩ \ Z) is defined by

u_{1} + g(u_{2}, …, u_{n}) = 0

where

g(u_{2}, …, u_{n}) = − log(−f(e^{u_{2}}, e^{u_{3}}, …, e^{u_{n}}))

for (u_{2}, … u_{n}) ∈ I_{2} × ⋯ I_{k} × I^{n−k}. Here I_{j} is a neighborhood of log |p_{j}|, 2 ≤ j ≤ k, and I = (−∞, −C) (C > 0 is a large constant).

The convexity of L(bΩ \ Z) implies that g(u_{2}, …, u_{n}) is a convex function. Therefore

ρ(z_{1}, …, z_{n}) = log |z_{1}| + g(log |z_{2}|, …, log |z_{n}|)
= log |z_{1}| − log(−f(|z_{2}|, …, |z_{n}|))

is a smooth plurisubharmonic defining function for bΩ near p.

is

formulate For z = (z_{1}, …, z_{n}), let ẑ = (z_{2}, …, z_{n}).

Theorem 3([B-S]). Suppose that a smooth hypersurface is defined by Re z_{1} + h(ẑ, Im z_{1}) = 0 near the origin. If h(ẑ, 0) ≥ 0, then the variety type of the hypersurface at the origin is obtained by analytic discs of form ϕ(ζ) = (0, ϕ_{2}(ζ), …, ϕ_{n}(ζ)). Furthermore, if the hypersurface is star-like at the origin, i.e., there exists a δ > 0 such that t → h(a_{2}t, …, a_{n}t, 0) is an increasing function on [0, δ] for all unit vectors (a_{2}, …, a_{n}), then the variety type at the origin is obtained by a complex line.

Now we can prove the main theorem of this paper:

Theorem 4. If Ω is a smoothly bounded pseudoconvex Reinhardt domain and if p ∈ bΩ, then the variety type at p is identical to the regular type.

Proof . It follows from the smoothness and the Reinhardt property of bΩ that p cannot be the origin (this can be easily seen from the proof of Lemma 2 in this paper or from Lemma 1.3 in [FIK]). We divide the proof into three cases.

If p ∈ bΩ \ Z, then by Lemma 1, L^{∗}(bΩ \ Z) is convex near p. Therefore, it follows from the results of McNeal [Mc] and Boas/Straube [B-S] that the variety type of L^{∗}(bΩ \ Z) at L^{∗}(p) is obtained by a complex line of the form L^{∗}(p) + aζ. Thus, the variety type of bΩ at p is obtained by

ϕ(ζ) = (p_{1}e^{a_{1}ζ}, …, p_{n}e^{a_{n}ζ}).

Therefore the variety type is identical to the regular type at p.

If p ∈ bΩ ∩ Z, then the mapping L^{∗} is not well-defined near p. Thus the above method fails. We first consider the case that only one coordinate of p is non-zero. We shall show that, in this case, the domain Ω is star-like at p. After a simple dilation and rotation, we may assume that p = (1, 0, …, 0) and Re z_{1} is the outward normal direction at p. It follows from the proof of Lemma 2 that bΩ is defined near p by

log |z_{1}| + h(ẑ) = 0

where h is a smooth function defined near p̂ such that (a) h(p̂) = 0 and dh(p̂) = 0;

(b) h(e^{iθ_{2}}z_{2}, …, e^{iθ_{n}}z_{n}) = h(z_{2}, …, z_{n});

(c) The function h is plurisubharmonic. Furthermore,

h(z_{2}, …, z_{n}) = g(log |z_{2}|, …, log |z_{n}|)

for some convex function g and z_{i} ≠ 0, 2 ≤ i ≤ n.

For a = (a_{2}, …, a_{n}) with ∥a∥ = 1 and a_{i} ≠ 0, 2 ≤ i ≤ n, define

k_{a}(t) = h(ta_{2}, …, ta_{n})
= g(log t + log |a_{2}|, …, log t + log |a_{n}|).

It follows from the convexity of g that g̃(s) = g(s + log |a_{2}|, …, s + log |a_{n}|) is a convex function of s. By (a), we have

lim_{s→−∞} g̃′(s) = 0.

Therefore, g̃′(s) ≥ 0 for s << −1. It then follows that k_{a}(t) is an increasing function of t. Thus, by the (proof of the) theorem of Boas/Straube, the variety type at p is obtained by a complex line of the form ϕ(ζ) = (1, b_{2}ζ, …, b_{n}ζ).

Now we treat the general situation for the case when p ∈ bΩ ∩ Z. Without lost of generality, we may assume that the first k coordinates of p are 1’s and the rest of the coordinates are 0’s. Let σ(z) be the Euclidean distance from z to bΩ. It follows from the proof of Lemma 2 that one of \frac{∂σ}{∂z_{j}} (p), 1 ≤ j ≤ k is not zero. For simplicity of notation, we assume that ∂σ(p)/∂z_{1} ≠ 0. Let z′ = Φ(z) be defined by

\begin{aligned}z′_{1} & = z_{1}^{α_{1}} ⋯ z_{k}^{α_{k}}; \\ z′_{j} & = z_{j}, 2 ≤ j ≤ n \end{aligned}

where α_{j} = ∂σ(p)/∂z_{j}, 1 ≤ j ≤ k.

It is obvious that z′ = Φ(z) is biholomorphic in a neigborhood of p and Re z′_{1} is the outward normal direction to bΩ at p. This change of coordinates preserves the Reinhardt property of the domain near p. Changing coordinates if necessary, we may assume at the outset that Re z_{1} is the outward normal direction to bΩ at p. It follows from the proof of Lemma 2 that bΩ is defined near p by

log |z_{1}| + h(z_{2}, …, z_{n}) = 0,

where h is a function satisfying properties (a)-(c). By a similar argument, we can prove that, for fixed z_{j}, 2 ≤ j ≤ k, near 1 and a = (a_{k+1}, …, a_{n}) with ∥a∥ = 1,

t ↦ h(z_{2}, …, z_{k}, ta)

is an increasing function of t for small t > 0. Thus we have

(2)  h(z_{2}, …, z_{n}) ≥ h(z_{2}, …, z_{k}, 0, …, 0).

Let ρ(w_{2}, …, w_{n}) = h(e^{w_{2}}, …, e^{w_{k}}, w_{k+1}, …, w_{n}). Then ρ(ŵ) is a plurisubharmonic function. Furthermore, for fixed (w_{k+1}, …, w_{n}), ρ(ŵ) is a convex function. Let k(t) = ρ(tw_{2}, …, tw_{k}, 0, …, 0). Then k(t) is a convex function with k(0) = k′(0) = 0. Therefore k(1) = ρ(w_{2}, …, w_{k}, 0, …, 0) ≥ 0. Hence h(ẑ) ≥ 0.

Claim. The variety type of bΩ at p is obtained by analytic discs of the form ϕ(ζ) = (1, ϕ̂(ζ)).

The proof of the claim comes directly from the proof of the theorem of Boas and Straube [B-S]. We provide details here for the reader’s convenience.

Let r(z) = log |z_{1}| + h(z_{2}, …, z_{n}). Let ϕ(ζ) = (ϕ_{1}(ζ), ϕ̂(ζ)): Δ → ℂ^{n} be an analytic disc such that ϕ(0) = p. We may assume that v(r ∘ϕ) > v(ϕ). If ϕ_{1}(ζ) is not identically zero, then there exists a sequence {ζ_{j}} ⊂ Δ, ζ_{j} → 0 such that {ϕ_{1}(ζ_{j})} is a sequence of real numbers and ϕ_{1}(ζ_{j}) → 1^{+}. Therefore

log |ϕ_{1}(ζ_{j})| = | log ϕ_{1}(ζ_{j})| ≤ r ∘ ϕ(ζ_{j}).

Let ψ(ζ) = (1, ϕ̂(ζ)). Since the order of vanishing of a holomorphic function is determined by its order of vanishing along a sequence, we obtain that

v(ϕ_{1}(ζ) − 1) = v(log |ϕ_{1}(ζ)|) ≥ v(r ∘ ϕ).

This in turn implies that v(ψ) = v(ϕ). It then follows from the equality

|r ∘ ψ(ζ) − r ∘ ϕ(ζ)| = | log |ϕ_{1}(ζ)||

that v(r ∘ ψ) ≥ v(r ∘ ϕ). This finishes the proof of the claim.

We now return to the proof of the main theorem. Let ϕ̃(ζ) = (ϕ_{2}(ζ), …, ϕ_{k}(ζ)) and let m = v(h ∘ ϕ̂). It follows from (2) that

(3)  v(h(ϕ̃(ζ), 0, …, 0)) ≥ m.

Let β_{j} = v(ϕ_{j}), k + 1 ≤ j ≤ n and ψ_{j}(ζ) = ϕ_{j}(ζ)/ζ^{β_{j}}. For δ > 0, we have

δ^{m} >_{∼} \frac{1}{2πδ} ∫_{|ζ|=δ} h ∘ ϕ̂(ζ)|dζ|

= \frac{1}{2πδ} ∫_{|ζ|=δ} h(ϕ̃(ζ), δ^{β_{k+1}}ψ_{k+1}(ζ), … δ^{β_{n}}ψ_{n}(ζ))|dζ|

≥ h(1, …, 1, δ^{β_{k+1}}ψ_{k+1}(0), … δ^{β_{n}}ψ_{n}(0)).

Therefore,

(4)  v(h(1, …, 1, ζ^{β_{k+1}}ψ_{k+1}(0), … ζ^{β_{n}}ψ_{n}(0))) ≥ m.

It follows from (3) and (4) that the variety type at p can be realized by analytic discs of the form

ϕ(ζ) = (1, ϕ_{2}(ζ), …, ϕ_{k}(ζ), 0, …, 0) or ϕ(ζ) = (1, …, 1, ϕ_{k+1}(ζ), …, ϕ_{n}(ζ)).

However, it follows from the first and second cases of this proof that the variety type can actually be realized by analytic discs of form

(5)  ϕ(ζ) = (1, e^{a_{2}ζ}, …, e^{a_{k}ζ}, 0, …, 0),

or

(6)  ϕ(ζ) = (1, …, 1, a_{k+1}ζ, …, a_{n}ζ).

Indeed, to obtain the special form (5), consider the section of Ω by Z_{k+1,...,n} = ⋂_{j=k+1}^{n} Z_{j}. Then, by Lemma 1.3 of [FIK], this section is a nonempty smoothly bounded set which is a finite collection of Reinhardt domains in Z_{k+1,...,n}. The point p then becomes a point with all nonzero components, and therefore the first case of the proof applies.

To obtain the special form (6), one needs to consider the section of Ω by a complex affine subspace of the form {z_{j_{1}} = ⋯ = z_{j_{k−1}} = 1}, where 1 ≤ j_{i} ≤ k for all i. Then we can use the argument from the second case of the proof.

Remark 5. (1) One certainly cannot expect that the line type at p is equal to the variety type. For example, let Ω be a pseudoconvex Reinhardt domain in ℂ^{2} such that bΩ is locally defined by

ρ(z_{1}, z_{2}) = log |z_{1}| + log |z_{2}| + (log |z_{1}| − log |z_{2}|)^{4}

for z = (z_{1}, z_{2}) near p = (1, 1). Then the line type at p is 2 while the regular type at p is 4.

(2) The result of Yu [Y] cannot be generalized to Reinhardt domains. Let Ω be a pseudoconvex Reinhardt domain defined by

ρ(z_{1}, z_{2}, z_{3}) = |z_{1}|^{2} + |z_{2}|^{6} + |z_{3}|^{6} + |z_{2}z_{3}|^{2} − 1.

Then the 3-tuple of the D’Angelo q-types at p = (1, 0, 0) is (Δ_{3}, Δ_{2}, Δ_{1}) = (1, 4, 6) while the Catlin multi-type is (1, 4, 4). We thank Professor Yu for pointing out this fact.

References

[B-S] H. Boas and E. Straube: On equality of line type and variety type of real submanifolds of ℂ^{n}, J. of Geometric Anal. 2 (1992), 95–98.

[FIK] S. Fu, A. Isaev and S. Krantz: Reinhardt domains with non-compact automorphism groups, preprint, 1995, 1–12.

[K] S. Krantz: Function Theory of Several Complex Variables, Second Ed., Wadsworth, Belmont, 1992.

[M] J. McNeal: Convex domains of finite type, J. of Functional Anal. 108 (1992), 361–373.

[Y] J. Yu: Multitypes of convex domains, Indiana Univ. Math. J., 41 (1992), 837–849.

Siqi Fu, Department of Mathematics, University of California, Riverside, CA 92521-0135, USA

  E-mail address: sfu@math.ucr.edu

A. V. Isaev, Centre for Mathematics and Its Applications, The Australian National University, Canberra, ACT 0200, Australia

  E-mail address: Alexander.Isaev@anu.edu.au

S. G. Krantz, Department of Mathematics, Washington University, St. Louis, MO 63130, USA

  E-mail address: sk@math.wustl.edu