PROPER HOLOMORPHIC MAPS FROM DOMAINS IN ℂ^{2} WITH TRANSVERSE CIRCLE ACTION

ADAM COFFMAN^{(1)} AND YIFEI PAN^{(2)}

Abstract. We consider proper holomorphic mappings between smoothly bounded pseudoconvex regions in complex 2-space, where the domain is of finite type and admits a transverse circle action. The main result is that the closure of each irreducible component of the branch locus of such a map intersects the boundary of the domain in the union of finitely many orbits of the group action.

1. Introduction

Let Ω be a bounded connected open subset of ℂ^{2}, with a smooth three-dimensional boundary bΩ, and assume that Ω is pseudoconvex and of finite type. Suppose further there is a continuous, non-constant homomorphism from the circle S^{1} to Aut(Ω), the Lie group of holomorphic automorphisms of Ω, with image denoted T, so we say Ω is a domain with T-action. If the T-action, with identity element e, has the following two properties:

(1) The group action T × Ω → Ω extends smoothly to T × \overline{Ω} → \overline{Ω};

(2) For each point z_{0} in bΩ, the evaluation map ψ_{z_{0}} : T → bΩ : θ ↦ θ(z_{0}) has differential map d(ψ_{z_{0}}) : T_{e}T → T_{z_{0}}bΩ with image not contained in the complex tangent line T_{z_{0}}^{h}bΩ;

then the T-action is called transverse ([Barrett]).

For a holomorphic map f : Ω → ℂ^{2}, let J_{f}(z) denote the Jacobian determinant of f at z ∈ Ω, and let V_{f} denote the branch locus, {z ∈ Ω : J_{f}(z) = 0}. The main result is:

Theorem 1.1. Given a smoothly bounded pseudoconvex finite type domain Ω ⊆ ℂ^{2} as above, with a transverse T-action, and another smoothly bounded pseudoconvex region D ⊆ ℂ^{2}, if f : Ω → D is a proper holomorphic map and W is an irreducible component of the analytic variety V_{f}, then the connected component of \overline{W} ∩ bΩ containing a point z_{0} is equal to the orbit ψ_{z_{0}}(T).

2000 Mathematics Subject Classification. 32H35, 32T25.

Key words and phrases. Proper holomorphic map, pseudoconvex domain.

We remark that this is similar to Theorem 1.1 of [CPS_{1}], which stated the additional hypothesis that D is a region of finite type, although in view of Proposition 2.2, this condition is superfluous. Our proof will be more elementary than that of [CPS_{1}] (modulo citation of well-known results of several complex variables from the 1980’s), and we take the opportunity to give a more detailed presentation. The methods also allow the statement, in Section 5, of a new, local version of the main result, where we formulate a notion of “locally transverse” and replace the condition “finite type at every point” on the boundary with finite type at a single point.

We further remark that several widely studied classes of domains admit a T-action: Hartogs domains, Reinhardt domains ([P]), and quasicircular domains ([Barrett], [CPS_{1}], [CPS_{2}]), but we will not consider any specific applications or examples here.

Returning to the statement of Theorem 1.1, the smooth, pseudoconvex, finite type properties of Ω imply that Ω satisfies (the well-known) Condition R ([D]), which implies that for any T-action, part (1) of the above definition of transverse is automatically satisfied ([Bedford_{3}]). Together with the hypotheses on Ω and D in Theorem 1.1, Condition R also implies ([Bell_{2}], [BC_{1}], [DF_{3}]) that the proper map f : Ω → D extends smoothly to the boundary, \overline{Ω} → \overline{D}. The Proof of Theorem 1.1 will proceed by analyzing the behavior of f on the boundary.

2. Geometry of the Domain

We begin by recalling some standard notions; for basic definitions and surveys of results regarding proper maps, see [Range], [Rudin] Chapter 15, [Bedford_{2}], [BN], [F], [D] Chapter 7.

A bounded domain Ω ⊆ ℂ^{2} with smooth boundary bΩ has a smooth (C^{∞}) global defining function r : ℂ^{2} → ℝ so that Ω = {r < 0}, bΩ = {r = 0}, and dr ≠ 0 on bΩ. Define the determinant of the Levi form, in terms of coordinates z_{1}, z_{2} on ℂ^{2}, to be the function Λ_{r} : ℂ^{2} → ℝ,

Λ_{r} = − det ( \begin{matrix}0 & rz̄_{1} & rz̄_{2} \\ r_{z_{1}} & r_{z_{1}z̄_{1}} & r_{z_{1}z̄_{2}} \\ r_{z_{2}} & r_{z_{2}z̄_{1}} & r_{z_{2}z̄_{2}} \end{matrix} ) .

Define (as in [Bell_{1}], [Bedford_{2}]) a function τ : bΩ → ℝ ∪ {+∞} by the rule that τ(p) is the order of vanishing of Λ_{r}|_{bΩ} at p (the smallest nonnegative integer m such that there is a tangential differential operator T of order m on bΩ such that T Λ_{r}(p) ≠ 0, or +∞ if there is no such m). The function τ does not depend on the choice of r. For pseudoconvex Ω, we denote the set of points at which Ω is not strictly Levi pseudoconvex by:

wbΩ = {z ∈ bΩ : Λ_{r}(z) = 0} = {z ∈ bΩ : τ(z) > 0}.

Definition 2.1. The condition that Ω is of “finite type” refers to the property that τ(p) is finite at each point p in bΩ. Equivalently ([D] §4.3.1), any complex curve in ℂ^{2} meets bΩ with at most a finite order of contact at each point.

Under the finite type assumption, τ : bΩ → [0, +∞) is upper semicontinuous: for each q ≥ 0, the set {p ∈ bΩ : τ(p) < q} is open, since its complement {z ∈ bΩ : T Λ_{r}(z) = 0 for all T with order < q} is closed, being the intersection over all T of closed zero sets of the smooth functions T Λ_{r}.

Proposition 2.2. Given a proper holomorphic map g : D_{1} → D_{2} between bounded pseudoconvex regions in ℂ^{2} with smooth boundaries, if g extends smoothly to bD_{1} in a neighborhood of p ∈ bD_{1}, then τ(p) ≥ τ(g(p)), and when τ(p) ≠ +∞, the following are equivalent:

• τ(p) = τ(g(p)),

• g extends to a local diffeomorphism at p,

• p ∉ \overline{V_{g}}.

Remark. A short proof is given in [Bell_{1}].

In particular, the Proposition applies to any automorphism θ ∈ T, which extends smoothly to \overline{Ω} by part (1) of the definition of transverse. So, for each z_{0} ∈ bΩ, τ is constant on the orbit ψ_{z_{0}}(T) containing z_{0}. Each orbit is a smoothly embedded circle in bΩ, by part (2) of the definition of transverse, and general properties of actions of Lie groups.

Lemma 2.3. Let Ω ⊆ ℂ^{2} be a smoothly bounded, pseudoconvex domain of finite type. Then, for any point p ∈ bΩ, there exists ϵ_{0} > 0 so that for any ϵ ∈ (0, ϵ_{0}), there is some δ ∈ (0, ϵ) with the following property: for any point q ∈ Ω ∩ B(p, δ), there exists a function h_{q} which is holomorphic on Ω, extends continuously to \overline{Ω}, and such that h_{q}(q) = 1, and |h_{q}(z)| ≤ \frac{1}{2} for all z ∈ Ω ∩ bB(p, ϵ).

Proof. Under the hypotheses of the Lemma, for any p ∈ bΩ, there exists a global peak function ([BF] Remark 3.4, [FM] Theorem 4.1), i.e., a function h_{p} with the following properties: h_{p} is continuous on \overline{Ω} and holomorphic on Ω, h_{p}(p) = 1, and |h_{p}(z)| < 1 on \overline{Ω} \ \{p\}. There is some ϵ_{0} > 0 so that bB(p, ϵ) ∩ \overline{Ω} ≠ Ø for any ϵ ∈ (0, ϵ_{0}). For any such ϵ, let M = max\{|h_{p}(z)| : z ∈ bB(p, ϵ) ∩ \overline{Ω}\} < 1. The set \{z ∈ \overline{Ω} : |h_{p}(z)| > M\} is an open neighborhood of p in \overline{Ω}, so there is some δ > 0 so that 0 ≤ M < |h_{p}(z)| for all z ∈ B(p, δ) ∩ Ω. For any q ∈ B(p, δ) ∩ Ω, let H_{q}(z) = h_{p}(z)/h_{p}(q), so H_{q}(q) = 1 and for z ∈ \overline{Ω} ∩ bB(p, ϵ), |H_{q}(z)| = |h_{p}(z)|/|h_{p}(q)| ≤ M/|h_{p}(q)| < 1. There is some integer N large enough so that h_{q}(z) = (H_{q}(z))^{N} has the properties claimed by the Lemma.

3. Geometry of the Branching Locus

First, for a proper map f as in Theorem 1.1, [Rudin] §15.1 shows J_{f} is not identically 0 on Ω, so V_{f} ≠ Ω. Further, it follows from Proposition 2.2 that for a proper map f as in Theorem 1.1, if p ∈ \overline{V_{f}} ∩ bΩ, then 0 ≤ τ(f(p)) < τ(p), so \overline{V_{f}} ∩ bΩ ⊆ wbΩ. From the well-known ([D],

[Rudin] §15.5) fact that for a bounded domain with smooth boundary Ω, wbΩ ≠ bΩ, we can conclude \overline{V_{f}} ∩bΩ ≠ bΩ (giving another proof that J_{f} ≢ 0).

In the case where Ω is a strictly pseudoconvex domain, it is known ([Bedford_{2}], [DF_{2}]) that f must be locally biholomorphic, so V_{f} is empty and Theorem 1.1 is vacuously true.

Another case where one containment from Theorem 1.1 is easy is when wbΩ is a union of only finitely many orbits. Then, since each orbit is a connected component of wbΩ, any connected component of any subset of wbΩ must be contained in exactly one orbit.

When V_{f} ≠ Ø, V_{f} is an analytic set in Ω of pure complex dimension 1, and has a unique decomposition into irreducible components ([GR]). Each component W is also an analytic set in Ω of pure complex dimension 1 which contains at least one smooth point of V_{f}.

For such a component W ⊆ Ω, its closure \overline{W} in ℂ^{2} meets the boundary bΩ in a closed set, E = \overline{W} ∩ bΩ. For a more precise description of how \overline{V_{f}} and \overline{W} meet bΩ, we will need the following Lemmas.

Various versions of the following result appear in [Bedford_{1}], [DF_{2}],

[P] Lemma 2.3, and [HJ] Lemma 3.2; we give a detailed proof here.

Lemma 3.1. Let Ω ⊆ ℂ^{2} be a smoothly bounded, pseudoconvex domain of finite type. Given a holomorphic map f : Ω → ℂ^{2} such that J_{f} ≢ 0 on Ω and f extends smoothly to bΩ, for an irreducible component W of the branch locus V_{f}, let E = \overline{W} ∩ bΩ. Then there is a dense subset S of E such that for each s ∈ S, there is a neighborhood U_{s} of s in ℂ^{2} and a smoothly embedded real 2-submanifold Y_{s} containing s in ℂ^{2} so that \overline{W} ∩ U_{s} = Y_{s} ∩ \overline{Ω} ∩ U_{s} and Y_{s} ∩ U_{s} meets bΩ transversely.

Proof. Consider the non-vacuous case where V_{f} ≠ Ø. Since J_{f} is holomorphic and ≢ 0 on Ω, there is some multi-index α so that the holo-

morphic function v = \frac{∂^{α}}{∂z^{α}}J_{f} is 0 on W, but the gradient ∇v is not identically 0 on W. We can assume (by switching coordinate labels if necessary) that there is some point w ∈ W where \frac{∂}{∂z_{1}} v(w) ≠ 0. Let W′ = \{z ∈ W : \frac{∂v}{∂z_{1}} (z) = 0\}, so (using the irreducibility assumption) W′ is a zero-dimensional subvariety of W.

By hypothesis, the function f extends to some smooth function on an open set U containing \overline{Ω}. Both v and \frac{∂v}{∂z_{1}} also extend smoothly to U, and v(p) = 0 for all p ∈ E by continuity.

Let S_{1} = \{z ∈ E : \frac{dv}{dz_{1}} (z) ≠ 0\}, so S_{1} is open in E. To show S_{1} is dense in E, it is enough to suppose, toward a contradiction, that there is some point p ∈ E and some neighborhood U_{0} of p in ℂ^{2} so that \frac{dv}{dz_{1}} is 0 at every point in U_{0} ∩ E. Corresponding to p, let ϵ_{0} > 0 be as in Lemma 2.3, so there is some ϵ < ϵ_{0} with \overline{B(p, ϵ)} ⊆ U_{0}, and a corresponding δ > 0. The set (W \ W′) ∩ B(p, δ) is a neighborhood of some point w in W with \frac{dv}{dz_{1}} (w) ≠ 0. Abbreviate B = B(p, ϵ) and bB = {|z − p| = ϵ}, so by Lemma 2.3, there is some holomorphic function h on Ω, extending continuously to \overline{Ω}, such that h(w) = 1 and |h(z)| ≤ \frac{1}{2} for all z ∈ Ω ∩ bB. Then, for any natural number N, the function (h(z))^{N} · \frac{dv}{dz_{1}} (z) is holomorphic when restricted to W ∩ B, and extends continuously to the closure \overline{W ∩ B}. It is elementary that \overline{W ∩ B} is contained in the union (W ∩ B) ∪ (W ∩ bB) ∪ (E ∩ \overline{B}), so h^{N} · \frac{dv}{dz_{1}} is identically 0 on one part of the boundary, E ∩ \overline{B}, and on the other part of the boundary, W ∩ bB, has magnitude less than |\frac{dv}{dz_{1}} (w)|, the magnitude of the function on an interior point, for sufficiently large N. This means the continuous, non-constant function |h^{N} · \frac{dv}{dz_{1}} | attains

its maximum value on the compact set \overline{W ∩ B} somewhere in W ∩ B, but this contradicts the maximum principle for holomorphic functions on the subvariety W ∩ B.

Continuing under the assumption that S_{1} ≠ Ø (since otherwise, E = Ø and the Lemma again holds vacuously), let q be some point in S_{1}, so v(q) = 0 and \frac{dv}{dz_{1}} (q) ≠ 0. This implies that for some neighborhood U_{q}^{1} of q in U, the extension of v to U restricts to ṽ_{q} on U_{q}^{1}, a smooth map U_{q}^{1} → ℂ ≅ ℝ^{2}, with real Jacobian of constant rank 2. The set Y_{q} = {z ∈ U_{q}^{1} : ṽ_{q}(z) = 0} is a smoothly embedded real surface in U_{q}^{1} ([Boothby] §III.5), with \overline{W} ∩ U_{q}^{1} ⊆ Y_{q} ∩ \overline{Ω}.

The tangent space T_{q}Y_{q} is a complex line (by continuity, since the tangent spaces are complex for points in Ω near q), so there is a complex affine coordinate transformation of ℂ^{2} taking q to the origin and T_{q}Y_{q} to the z_{1}-axis. Then there is some open polydisc U_{q}^{2} = D_{q}^{2}×H_{q}^{2} centered at q and contained in the (transformed) U_{q}^{1} neighborhood so that the intersection of Y_{q} with U_{q}^{2} is a graph over the disc D_{q}^{2} × {0} = T_{q}Y_{q} ∩ U_{q}^{2}. In the transformed coordinates, Y_{q} ∩ U_{q}^{2} = {(z_{1}, g_{q}(z_{1})) : z_{1} ∈ D_{q}^{2}}, for some smooth function g_{q} : D_{q}^{2} → H_{q}^{2}. Denote the graphing diffeomorphism Γ_{q} : D_{q}^{2} → U_{q}^{2} : z_{1} ↦ (z_{1}, g_{q}(z_{1})). The set Γ_{q}^{−1}(Ω ∩ U_{q}^{2}) = {z_{1} ∈ D_{q}^{2} : (z_{1}, g_{q}(z_{1})) ∈ Ω} is open in D_{q}^{2}, and g_{q} and Γ_{q} are holomorphic on this set.

Let ρ_{q} : ℂ^{2} → ℝ be a smooth defining function for the domain Ω = {z : ρ_{q}(z) < 0}, chosen so that −(−ρ_{q})^{2/3} is strictly plurisubharmonic on Ω ∩ U_{q}^{3}, where U_{q}^{3} is a neighborhood of q in ℂ^{2} ([DF_{1}]). We can assume, by intersecting and shrinking if necessary, that U_{q}^{3} is a polydisc D_{q}^{3} × H_{q}^{3} centered at q and contained in U_{q}^{2}.

Define σ_{q} : D_{q}^{3} → ℝ : σ_{q}(z_{1}) = ρ_{q}(z_{1}, g_{q}(z_{1})), so σ_{q} = ρ_{q} ∘ Γ_{q}|_{D_{q}^{3}} is smooth and σ_{q}(z_{1}) = 0 on the set {z_{1} ∈ D_{q}^{3} : (z_{1}, g_{q}(z_{1})) ∈ bΩ}. The real-valued function ϕ_{q}(z_{1}) = −(−σ_{q}(z_{1}))^{2/3} is continuous on D_{q}^{3}, and plurisubharmonic, and therefore subharmonic, on the open subset {z_{1} ∈ D_{q}^{3} : (z_{1}, g_{q}(z_{1})) ∈ Ω ∩ U_{q}^{3}}, being a composite of the smooth, strictly plurisubharmonic function −(−ρ_{q})^{2/3} on Ω ∩ U_{q}^{3} with a holomorphic map ([Range] Corollary II.4.10).

Consider the partition of D_{q}^{3} into the open set {σ_{q} < 0} and the closed set {σ_{q} ≥ 0} with common boundary Σ_{q}. It is elementary that the set of points s in D_{q}^{3} such that there exists an open disc Δ_{s} with \overline{Δ_{s}} ⊆ D_{q}^{3} and \overline{Δ_{s}} ∩ \{σ_{q} ≥ 0\} = \{s\} is a dense subset of Σ_{q}. On such a disc \overline{Δ_{s}}, the function ϕ_{q} attains its maximum value, 0, at a unique point, s. Let ν be the outward unit normal vector field on the boundary of Δ_{s}, so ν_{s} is the unit normal vector at s. By the Hopf Lemma ([Range] §II.4), there is some C > 0 so that ϕ_{q}(s−xν_{s}) ≤ −Cx for x in some interval (0, x_{0}). It follows that σ_{q}(s − xν_{s}) ≤ −(Cx)^{3/2}, and since σ_{q} is smooth, this is enough to show the normal derivative at s, ν_{s}(σ_{q}), is non-zero. By construction, the pushforward vector field (Γ_{q})_{∗}ν on the boundary of Γ_{q}(Δ_{s}) ⊆ Y_{q}, when applied to the function ρ_{q} at (s, g_{q}(s)) ∈ E, gives

((Γ_{q})_{∗}ν)_{Γ_{q}(s)}(ρ_{q}) = ν_{s}(ρ_{q} ∘ Γ_{q}) = ν_{s}(σ_{q}) ≠ 0.

This shows Y_{q} meets bΩ transversely at (s, g_{q}(s)), and such points form a dense subset of Γ_{q}(Σ_{q}) = E ∩ U_{q}^{3}.

The conclusion is that for any q ∈ S_{1}, Y_{q}∩U_{q}^{3} meets bΩ transversely at a point in E arbitrarily close to q, and since S_{1} is open and dense in E, S is also dense in E. Since transverse intersection is an open condition, Y_{q} meets bΩ transversely in some neighborhood of each point in S.

Lemma 3.2. Let Ω ⊆ ℂ^{2} be a smoothly bounded domain. Given a holomorphic map f : Ω → ℂ^{2} such that J_{f} ≢ 0 on Ω and f extends smoothly to bΩ, with branch locus V_{f} ⊆ Ω, the set \overline{V_{f}} ∩ bΩ does not contain any totally real surface.

Proof. A “totally real surface” is a smoothly embedded real surface in ℂ^{2} such that none of its tangent planes is a complex line. We remark that this Lemma does not follow from Lemma 3.1 — if we write the branch locus as a union of irreducible components V_{f} = ∪W_{α}, we could immediately conclude \overline{V_{f}} ⊇ ∪\overline{W_{α}}, but this is not enough. Instead, we recall a result of Pinčuk ([Pinčuk], [Rudin], §10.6), which states that if the holomorphic function J_{f} extends continuously to bΩ and vanishes identically on some totally real surface in bΩ, then J_{f} ≡ 0 on Ω.

4. The Main Result

Recall the main result, Theorem 1.1, states that the connected component of E = \overline{W} ∩ bΩ containing a point z_{0} is equal to the orbit ψ_{z_{0}}(T). By the definition of connected, the equality of sets will follow from showing the set E ∩ ψ_{z_{0}}(T) is both closed and open in both E and ψ_{z_{0}}(T). Since E is closed and the orbit is an embedded circle, the intersection is closed. The remainder of this Section will show the intersection is open in both E and ψ_{z_{0}}(T).

Theorem 4.1. For Ω, D, f, W as in Theorem 1.1, let E = \overline{W} ∩ bΩ. For any point p in E, and any neighborhood U of p in ℂ^{2}, there is some ϵ > 0, a point v ∈ E∩U, and a smoothly embedded arc γ_{p} : (−ϵ, ϵ) → ℂ^{2} with γ_{p}(0) = v and γ_{p}((−ϵ, ϵ)) ⊆ E ∩ ψ_{v}(T) ∩ U.

Proof. By Lemma 3.1, there is some s ∈ S ∩ U, some neighborhood U_{s} ⊆ U of s, and some smooth manifold Y_{s} meeting bΩ transversely in U_{s} so Y_{s}∩bΩ∩U_{s} = E∩U_{s}. In U_{s}∩bΩ, the function τ is positive on E and attains its minimum positive value at some point v ∈ E ∩ U_{s}. Because τ is upper semicontinuous, the set U_{v} = {z ∈ U_{s} ∩bΩ : τ(z) < τ(v)+1} is open, and so the set where τ(z) has constant value τ(v) is an open neighborhood of v in E ∩ U_{s}. Because E is a transverse intersection in U_{s}, it is a smooth curve, admitting a local parametrization γ_{p} : (−ϵ_{1}, ϵ_{1}) → E ∩ U_{v}, with γ_{p}(0) = v and nonvanishing velocity vector γ′_{p}(t).

Consider a particular point z_{0} = γ_{p}(t_{0}) in E ∩U_{v}, which is also on its orbit curve under the group action on \overline{Ω}, so z_{0} = ψ_{z_{0}}(e). Both tangent vectors γ′_{p}(t_{0}) and ψ′_{z_{0}}(e) are non-zero; suppose toward a contradiction that they are linearly independent. It is then elementary that the group action (−ϵ_{1}, ϵ_{1}) × T → bΩ : (t, θ) ↦ ψ_{γ_{p}(t)}(θ) parametrizes an embedded surface patch, for some neighborhood of (t_{0}, e) in (−ϵ_{1}, ϵ_{1})× T ([Boothby] Ch. III). This surface P in bΩ is totally real in ℂ^{2}; by part (2) of the transverse group action hypothesis, a tangent vector to an orbit is not contained in any complex line tangent to bΩ. Further, P contains z_{0}, and meets E in some open arc in γ_{p}((−ϵ_{1}, ϵ_{1})). Since τ is constant on the arc and on each orbit curve, τ has the constant value τ(z_{0}) on P.

By Proposition 2.2, τ(f(z_{0})) < τ(z_{0}), while τ(f(z)) = τ(z) for all z ∉ \overline{V_{f}}. Using the upper semicontinuity of τ on bD, the set \{w ∈ bD : τ(w) < τ(f(z_{0})) + 1} is open in bD, so its inverse image under f, {z ∈ bΩ : τ(f(z)) ≤ τ(f(z_{0}))}, is open in bΩ, and contains z_{0}, but no points q in P \ \overline{V_{f}}, which would have the property τ(f(q)) = τ(q) = τ(z_{0}) > τ(f(z_{0})). The conclusion is that there is some totally real surface patch, a neighborhood of z_{0} in P, contained in \overline{V_{f}}. However, this contradicts Lemma 3.2.

So, for each t in an interval around 0, the velocity vector γ′_{p}(t) is tangent to some orbit ψ_{γ_{p}(t)}(θ), meaning the arc γ_{p} is an integral curve of the line field in bΩ tangent to the orbits. Therefore, the arc coincides with the single orbit ψ_{v}(θ) in some neighborhood of v = γ_{p}(0) (a simple case of the Frobenius Theorem, [Boothby] §IV.8).

We need one more fact about the geometry of Ω.

Lemma 4.2. For Ω as in Theorem 1.1, and any point p ∈ bΩ, let O denote the orbit ψ_{p}(T). Then there exists a neighborhood U_{O} of O in ℂ^{2}, and a smooth analytic variety A_{O} in U_{O} ∩Ω such that \overline{A_{O}} ∩bΩ = O.

Proof. We begin by transforming a neighborhood of p in bΩ into a “rigid” hypersurface, using the hypotheses that Ω has smooth boundary and admits a transverse T-action. In particular, bΩ, considered as an abstract CR manifold, has a CR structure invariant under the transverse action of T, and by a result of [BRT], there exists a neighborhood U_{p} of p in ℂ^{2}, and a smooth CR embedding F̃ : bΩ∩U_{p} → ℂ^{2}, so that in the target coordinate system (z̃_{1}, z̃_{2}), z̃_{j} = x̃_{j} + iỹ_{j}, F̃(p) = 0 and the image of the embedding is defined by the smooth real graphing equation ỹ_{2} = H(x̃_{1}, ỹ_{1}) = H(z̃_{1}) in some neighborhood of the form V = {(z̃_{1}, z̃_{2}) : |z̃_{1}| < η, |x̃_{2}| < r_{1}, |ỹ_{2}| < r_{2}}. The main property is that H does not depend on x̃_{2}. Further, for q ∈ bΩ ∩ U_{p}, the map F̃ takes orbits O_{q} = ψ_{q}(T) to real lines parallel to the x̃_{2}-axis: F̃(O_{q} ∩ U_{p}) ∩ V = {(z̃_{1}, z̃_{2}) ∈ V : z̃_{1} = c_{q}, |x̃_{2}| < r_{1}, ỹ_{2} = H(c_{q})}, and in the special case q = p, denoting O = O_{p}, F̃(O ∩ U_{p}) ∩ V is the x̃_{2}-axis, i.e., {ỹ_{2} = z̃_{1} = 0}. (See also the survey [C], or [T] for the real analytic case.)

Since bΩ is a pseudoconvex hypersurface, and the Levi form is invariant under the CR isomorphism F̃, the image hypersurface is also pseudoconvex. By a result of [BC_{2}] (which also uses the hypothesis that Ω is of finite type), F̃ extends to a one to one map F : U_{p} → ℂ^{2} (possibly shrinking the neighborhood U_{p} of p) with the following properties. The map F is biholomorphic on Ω∩U_{p}, extends smoothly to \overline{Ω}∩U_{p}, and agrees with the diffeomorphism F̃ on bΩ ∩ U_{p}. In terms of the smooth graphing function H, (and, again possibly shrinking the polydisc V), the map F has the property F (Ω ∩ U_{p}) ∩ V = {H(z̃_{1}) < ỹ_{2} < r_{2}}.

As a remark which will not be needed until later, we note that given a set V, the radius η > 0 can be shrunk arbitrarily while r_{1} and r_{2} are fixed, and the thinner neighborhood will still meet F (bΩ ∩ U_{p}) in a graph with the claimed properties.

For each complex number c in the disc {|z̃_{1}| < η}, let A_{c} denote the set {z̃_{1} = c, |x̃_{2}| < r_{1}, H(c) < ỹ_{2} < r_{2}}, which is a smooth, irreducible analytic variety in the open set F (Ω∩U_{p})∩V. It has the property that \overline{A_{c}} ∩ F (bΩ ∩ U_{p}) ∩ V is a line segment, \{z̃_{1} = c, |x̃_{2}| < r_{1}, ỹ_{2} = H(c)\}, equal to the image of some orbit, F (O_{q} ∩ U_{p}) ∩ V.

In fact, A_{c} is the unique irreducible analytic variety in F (Ω∩U_{p})∩V meeting F (bΩ∩U_{p})∩V along F (O_{q}∩U_{p})∩V. Without loss of generality, consider the particular case c = 0, where A_{0} is the intersection of the z̃_{2}-axis with F (Ω∩U_{p})∩V, and \overline{A_{0}} ∩F (bΩ∩U_{p})∩V = F (O ∩U_{p})∩V is the x̃_{2}-axis in V. Suppose there is some point q̃ ∈ F (O ∩ U_{p}) ∩ V, and some neighborhood V′ of q̃ in V, and some irreducible one-dimensional analytic variety A′ in F (Ω ∩ U_{p}) ∩ V′ with closure \overline{A′} in V′ so that \overline{A′} ∩ F (bΩ ∩ U_{p}) ⊆ F (O ∩ U_{p}). To show A′ ⊆ A_{0}, we will use a maximum principle argument similar to that from the Proof of Lemma 3.1. The function z̃_{1} is holomorphic on A′, and extends continuously to \overline{A′}, where it is identically 0 on \overline{A′} ∩ F (bΩ ∩ U_{p}). If A′ is not contained in A_{0}, then the set of points in A′ with coordinates (z̃_{1}, z̃_{2}), z̃_{1} = 0, is zero-dimensional in A′. Corresponding to p′ = F^{−1}(q̃) in bΩ, let ϵ_{0} > 0 be as in Lemma 2.3, so there is some ϵ < ϵ_{0} with F (\overline{B(p′, ϵ)}) ⊆ V′, and a corresponding δ > 0. The set A′ ∩ F (B(p′, δ)) contains some point w̃ = (w̃_{1}, w̃_{2}), with w̃_{1} ≠ 0. Abbreviate B = B(p′, ϵ) and bB = {|z − p′| = ϵ}, so by Lemma 2.3, there is some holomorphic function h on Ω, extending continuously to \overline{Ω}, such that h(F^{−1}(w̃)) = 1 and |h(z)| ≤ \frac{1}{2} for all z ∈ Ω ∩ bB. Then, for any natural number N, the function ((h ∘ F^{−1})(z̃_{1}, z̃_{2}))^{N} · z̃_{1} is holomorphic when restricted to A′ ∩ F (B), and extends continuously to the closure \overline{A′ ∩ F (B)}. Since \overline{A′ ∩ F (B)} is contained in the union (A′ ∩F (B))∪(A′ ∩F (bB))∪((\overline{A′} ∩ F (bΩ ∩ U_{p})) ∩ \overline{F (B)}), (h ∘ F^{−1})^{N} · z̃_{1} is identically 0 on one part of the boundary, \overline{A′} ∩ F (bΩ ∩ U_{p}) ∩ \overline{F (B)}, and on the other part of the boundary, A′ ∩ F (bB), has magnitude less than |w̃_{1}|, the magnitude of the function on an interior point, for sufficiently large N. This means the continuous, non-constant function |(h ∘ F^{−1})^{N} · z̃_{1}| attains its maximum value on the compact set \overline{A′ ∩ F (B)} somewhere in A′ ∩ F (B), but this contradicts the maximum principle for holomorphic functions on the subvariety A′ ∩ F (B).

Since this construction can be carried out at any point p on O, O is covered by neighborhoods F^{−1}(V), so that the open sets F^{−1}(V) ∩ Ω contain analytic sets F^{−1}(A_{0}) that patch together smoothly on overlaps near bΩ by the above uniqueness. By the compactness of O, it is covered by finitely many such neighborhoods F^{−1}(V), whose union contains a neighborhood U_{O} of O, small enough so that its intersection with the union of varieties F^{−1}(A_{0}) is a closed variety A in U_{O} ∩ Ω.

At this point we have enough to establish the main result:

Proof of Theorem 1.1. Given z_{0} ∈ \overline{W} ∩ bΩ, let E = \overline{W} ∩ bΩ, and let E_{0} denote the connected component of E containing z_{0}.

Apply Lemma 4.2 to z_{0}, to get a one-sided holomorphic map F : U_{z_{0}} → ℂ^{2}, and a neighborhood V ⊆ F (U_{z_{0}}) centered at 0 = F (z_{0}) and meeting F (U_{z_{0}} ∩ bΩ) in a graph as in the Proof of Lemma 4.2. Fixing the lengths r_{1}, r_{2} of V, consider any small η > 0 and a subset V_{η} = {(z̃_{1}, z̃_{2}) ∈ V : |z̃_{1}| < η} as in the remark from the Proof of Lemma 4.2. From Theorem 4.1 applied to the neighborhood F^{−1}(V_{η}) of z_{0}, there is some v ∈ F^{−1}(V_{η}) and an arc γ((−ϵ, ϵ)) inside E ∩ ψ_{v}(T) ∩ F^{−1}(V_{η}). The composite F ∘ γ parametrizes a segment parallel to the x̃_{2}-axis in V_{η}, and part of the boundary of some irreducible variety A_{c} with |c| < η. However, this segment is also part of the boundary of the variety F (W ∩ U_{z_{0}}), and by the uniqueness from the Proof of Lemma 4.2, the varieties A_{c} and F (W ∩U_{z_{0}}) meet in some open subset of A_{c}, so A_{c} ⊆ F (W ∩U_{z_{0}}). Since η can be arbitrarily small, it follows that every point on A_{0} is a limit point of F (W ∩U_{z_{0}}), being within η of some point in A_{c}, and since F (W ∩ U_{z_{0}}) is closed in F (U_{z_{0}}), A_{0} ⊆ F (W ∩ U_{z_{0}}).

This shows that any point z_{0} ∈ E is contained in some arc in E ∩ ψ_{z_{0}}(T), which is open in ψ_{z_{0}}(T). Since E is closed, E ∩ ψ_{z_{0}}(T) is both open and closed in ψ_{z_{0}}(T), and since the orbit is connected, ψ_{z_{0}}(T) ⊆ E_{0}.

To show ψ_{z_{0}}(T) is open in E, which as previously mentioned will complete the proof, it is enough to show that any point z_{0} ∈ E has a neighborhood U in ℂ^{2} so that U ∩ E ⊆ ψ_{z_{0}}(T). Since this is a local statement, we again consider a transformation F and coordinate neighborhood V as in Lemma 4.2, and we need to show that there is some neighborhood of 0 ∈ V that meets F (E ∩ U_{z_{0}}) ∩ V only in the x̃_{2}-axis.

The point (z̃_{1}, z̃_{2}) = (0, \frac{i}{2}r_{2}) is in the open set F (U_{z_{0}} ∩ Ω), so it is the center of some polydisc

P_{I} = \{|z̃_{1}| < δ_{1}, |z̃_{2} − \frac{i}{2}r_{2}| < ρ\} ⊆ F (U_{z_{0}} ∩ Ω) ∩ V.

For 0 < δ ≤ δ_{1}, denote another polydisc

P_{δ} = {|z̃_{1}| < δ, |z̃_{2}| < ρ} ⊆ V,

centered at 0. Suppose, toward a contradiction, that for any δ, there is some point w̃ = (w̃_{1}, w̃_{2}) ∈ F (E ∩ U_{z_{0}}) ∩ P_{δ} with w̃_{1} ≠ 0. Then, by the above argument, Aw̃_{1} = {z̃_{1} = w̃_{1}, |x̃_{2}| < r_{1}, H(w̃_{1}) < ỹ_{2} < r_{2}} is a subset of F (W ∩ U_{z_{0}}) ∩ V. It follows that for any complex number ξ in the disc \{|ξ − \frac{i}{2}r_{2}| < ρ\}, the point (w̃_{1}, ξ) is an element of P_{I} ∩ F (W ∩ U_{z_{0}}), and since δ can be arbitrarily small, (0, ξ) is an accumulation point of the analytic set {z̃_{2} = ξ}∩P_{I} ∩F (W ∩U_{z_{0}}) in the radius δ_{1} disc D_{ξ} = {z̃_{2} = ξ}∩P_{I}. This implies each disc D_{ξ} is a subset of F (W ∩ U_{z_{0}}), so the complex analytic variety F (W ∩ U_{z_{0}}) contains the open set P_{I}, contradicting the fact that W is one-dimensional.

Corollary 4.3. For W as in Theorem 1.1, \overline{W} ∩ bΩ is a finite union of orbits, and in a neighborhood of bΩ, the set \overline{W} is a submanifold with boundary.

Proof. It follows from the construction in the above Proof that at each point z_{0} ∈ \overline{W} ∩ bΩ, there is some neighborhood of z_{0} where \overline{W} ∩ bΩ meets exactly one orbit, and by compactness, \overline{W} ∩ bΩ can be covered by finitely many such neighborhoods. The claim about \overline{W} being a submanifold with boundary near bΩ, in fact coinciding with some variety A_{O} near each boundary component, also follows from the observations in the previous Proof.

5. A Local Version

The above method of proof of the main result allows for a local version, which improves on a claim of [CPS_{2}], its Theorem 2, which was stated without a complete proof.

On a smoothly bounded domain, if a point on the boundary is of finite type, then by upper semicontinuity, it has a neighborhood of points of finite type. Most of the above steps — including the transverse intersection Lemma 3.1, the arc intersections in Theorem 4.1, the transformation to rigid form in Lemma 4.2, and the topological and geometric arguments in the proof of Theorem 1.1 — are stated in local terms. The only use of a global object was in Lemma 2.3, where, in the interest of simplicity, we cited a result on the existence of a global peak function. However, we only need to apply the conclusion of Lemma 2.3 locally, and (as explained to the authors by G. Bharali) the method of [BF] in fact gives a local peak function at a point of finite type in the boundary of a pseudoconvex domain, so the Proof of Lemma 2.3 can be adapted to give the following:

Lemma 5.1. Let Ω ⊆ ℂ^{2} be a smoothly bounded pseudoconvex domain, and let p be a point of finite type in the boundary. Then there exists ϵ_{0} > 0 so that for any ϵ ∈ (0, ϵ_{0}), there is some δ ∈ (0, ϵ) with the following property: for any point q ∈ Ω∩B(p, δ), there exists a function h_{q} which is holomorphic on Ω ∩ B(p, ϵ), extends continuously to \overline{Ω} ∩ \overline{B(p, ϵ)}, and such that h_{q}(q) = 1, and |h_{q}(z)| ≤ \frac{1}{2} for all z ∈ Ω ∩ bB(p, ϵ).

Using this version of the Lemma, but otherwise not significantly changing the argument, allows a modification of the main result. We continue to consider a domain Ω in ℂ^{2} with a (global) T-action, but which is not necessarily transverse everywhere. We also make the assumption that Ω satisfies Condition R. Recall Condition R implies part (1) of the definition of transverse T-action, and conversely, it was shown by [Barrett] that if the T-action is (globally) transverse, then the domain satisfies Condition R. We consider a particular point z_{0} in the boundary where d(ψ_{z_{0}}) : T_{e}T → T_{z_{0}}bΩ has image not contained in the complex tangent line T_{z_{0}}^{h}bΩ; this property is shared by points in a neighborhood of z_{0}, and since T acts by automorphisms, the property holds in a neighborhood of the orbit ψ_{z_{0}}(T), and we say that the T-action is locally transverse at z_{0}.

If there exists a proper map f : Ω → D, and D is pseudoconvex, then Ω is also pseudoconvex ([Range] §II.5). So, as before, it follows that any proper map from Ω to a smooth, pseudoconvex region D extends smoothly to the boundary.

Corollary 5.2. Given a smoothly bounded domain Ω in ℂ^{2} satisfying Condition R and admitting a T-action, suppose z_{0} ∈ bΩ is a point of finite type where the T-action is locally transverse. Let D ⊆ ℂ^{2} be a smoothly bounded pseudoconvex region. If f : Ω → D is a proper holomorphic map and W is an irreducible component of the analytic variety V_{f} such that z_{0} ∈ \overline{W} ∩ bΩ, then the connected component of \overline{W} ∩bΩ containing z_{0} is equal to the orbit ψ_{z_{0}}(T), and in a neighborhood of the orbit, \overline{W} is a submanifold with boundary.

Acknowledgments. The authors thank G. Bharali for helpful conversations regarding Lemmas 2.3 and 5.1.

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(1,2)Department of Mathematical Sciences, Indiana University - Purdue University Fort Wayne, Fort Wayne, IN 46805-1499

(2)College of Mathematics and Information Sciences, Jiangxi Normal University, Nanchang, China

E-mail address: CoffmanA@ipfw.edu, Pan@ipfw.edu