Automorphism group actions in complex analysis

Journal Pre-proof Automorphism group actions in complex analysis Steven G. Krantz

PII: DOI: Reference:

S0723-0869(19)30033-7 https://doi.org/10.1016/j.exmath.2019.09.001 EXMATH 25376

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Expositiones Mathematicae

Received date : 26 February 2019 Accepted date : 23 September 2019 Please cite this article as: S.G. Krantz, Automorphism group actions in complex analysis, Expositiones Mathematicae (2019), doi: https://doi.org/10.1016/j.exmath.2019.09.001. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2019 Elsevier GmbH. All rights reserved. ⃝

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Automorphism Group Actions in Complex Analysis12 by Steven G. Krantz

February 26, 2019

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Abstract: In this paper we describe the subject of automorphism groups of domains in complex space. This has been an active area of research for fifty years or more, and continues to be dynamic and developing today. We discuss noncompact automorphism groups, the Bun Wong/Rosay theorem, the Greene/Krantz conjecture, semicontinuity of automorphism groups, the method of scaling, and other current topics. Contributions from geometers, Lie theorists, analysts, and function theorists are described.

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Background

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In this paper a domain in Euclidean space is a connected open set. Generally we shall restrict attention to bounded domains, and typically our domains will have smooth boundary. It should be noted immediately that this makes our discussion distinct from the study of the bounded symmetric domains ´ Cartan (see [HEL, Ch. III]). Of his four types of domains (and two of E. exceptional domains), only one (the ball) has smooth boundary. The others have Lipschitz boundaries. Throughout this paper, we let D denote the unit disc in the complex plane and Ω denote a smoothly bounded domain in any dimensional complex space. Usually it is convenient to think of a domain Ω as given by a defining function. Here a defining function ρ = ρΩ for Ω has the properties that Ω = {z ∈ Cn : ρ(z) < 0}

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AMS Subject Classification Numbers: 32M05, 32M17, 32H02, 32M25, 32M99 Key Words: domain, automorphism group, isotropy group, boundary orbit accumulation point, pseudoconvex 2

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and ∇ρ 6= 0 on ∂Ω. In case the defining function ρ is C k then we say that ∂Ω has C k boundary. This definition also applies to C ∞ boundary and C ω boundary. These definitions of smooth boundary are equivalent to other standard definitions (see Appendix I in [KRA1] for the details). The automorphism group of Ω is the collection of biholomorphic maps from Ω to itself. Such a mapping is of course one-to-one and onto with a holomorphic inverse. We denote the automorphism group by Aut (Ω). The binary operation on the automorphism group is composition of mappings. The natural topology on the automorphism group is that of uniform convergence on compact sets (otherwise known as the compact-open topology).3 For a bounded domain Ω in Cn , Aut (Ω) is a real Lie group but never a complex Lie group, unless the group is zero dimensional (see [KOB1] for the details). It is unfortunately the generic case that a domain in Cn will have only trivial automorphisms group, i.e., automorphism group equal to the identity only (see [BSW] and [GRK]). It is natural to consider domains Ω with transitive automorphism group. The disc in the plane and the ball in complex space both have this property. But it turns out that, in a fairly natural sense, these are the only smoothly bounded domains with transitive automorphism group (see the details below). For this reason we expand our horizons to a larger and geometrically very natural class of domains called “domains with noncompact automorphism group.” A domain Ω has this last property if the automorphism group is not compact in the topology described in the preceding paragraph. Using a classical result of H. Cartan, we can also characterize domains with noncompact automorphism group by this property (see [NAR]): There exists a point P ∈ Ω and a point X ∈ ∂Ω and a sequence {ϕj } of automorphisms of Ω so that ϕj (P ) → X as j → ∞. We shall make extensive use of this last formulation in the discussions that follow.

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Introduction to Automorphism Groups

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A celebrated result of Poincar´e in 1906 tells us that the ball and the bidisc in C2 are not biholomorphic.4 As a result, we see that there is no Riemann mapping theorem in the context of several complex variables. More recent 3

Let ϕj , ϕ0 ∈ Aut (Ω). When ϕj → ϕ0 uniformly on compact sets then we shall say that ϕj converges normally to ϕ0 . 4 In fact his proof consists in comparing the automorphism groups of the domains.

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results of Greene/Krantz [GRK1], [GRK3], Burns/Shnider/Well [BSW], and others illustrate how decisively it is the case that, in a generic sense, the ball in Cn is not biholomorphic to most any other topologically trivial domain in Cn . On the other hand, the notable boundary regularity result of Fefferman [FEF1] makes possible (at least in principle) the program of Poincar´e to classify domains in Cn up to biholomorphic equivalence. The paper [CHM], inspired in part by [FEF1], actually lays the foundations for that program on strongly pseudoconvex domains and more generally on domains with nondegenerate Levi form. Others, including Moser/Webster [MOW] and Gong [GON], have taken some of the first steps in that program. In fact there has been a great deal of work in the past forty years in understanding and fleshing out Poincar´e’s program. More will be said about this matter in what follows. As a result of these considerations, there is some interest in finding substitutes for the Riemann mapping theorem and for the classification theory of domains in several complex variables. One such ersatz is the study of the automorphism group of a domain.5 This group is a convenient biholomorphic invariant that allows us to distinguish and compare domains in a productive fashion. The classical theory of automorphisms consisted largely of calculating the automorphism groups of specific domains. Such an enterprise is quite limited, as the automorphism group of a domain can rarely be ´ Cartan specified explicitly. Beyond the bounded symmetric domains of E. (see [HEL]) there is little that can be said. But the modern theory, which can be said to have begun in the 1970s, considers entire classes of smoothly bounded domains. It addresses new types of results that are geometrically compelling. There are several important new directions in the most recent work. Certainly investigations of the Greene/Krantz conjecture (to be explained below) play a notable role. Variants of the original Bun Wong/Rosay theorem appear repeatedly. Semicontinuity of automorphism groups has been a fruitful topic. Characterizations of domains by their automorphism groups has been a new direction of research. The role of differential geometry is becoming ever more prominent. The technique of scaling is a major tool. We shall devote considerable space below to all these developments. Just to set the stage for what is to follow, we consider a simple example 5 Although, to be fair, most domains have trivial automorphism group. So the approach suggested here has its limitations.

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of the idea of characterizing a domain by its automorphism group. It is not difficult to see that domains with “small” automorphism groups are not amenable to simple characterizations. EXAMPLE 2.1 Let G be the group consisting only of the identity element. In a palpable sense, most strongly pseudoconvex domains have automorphism group G (see [GRK1]); and strongly pseudoconvex domains are generically biholomorphically distinct from each other (see [GRK1], [GRK3], [BSW]). So this is a situation in which we cannot hope to classify domains by their automorphism group.

The Situation in Complex Dimension One

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It seems likely that the first result we shall present has been known to Riemann surface theorists for some time. But no reference is apparent in the literature. The theorem does appear in the paper [KRA2]. We begin with a preliminary result which has intrinsic interest but which will also be needed in the arguments below.

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Theorem 3.1 Let {ϕj } be a sequence of automorphisms of a bounded domain Ω ⊆ C that converges uniformly on compact sets. Then the limit function is either itself an automorphism or else it is constantly equal to a point of ∂Ω.

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Proof: Of course the limit function ϕ0 will be holomorphic. Now we examine the sequence ψj ≡ ϕ−1 j . These form a bounded sequence of holomorphic functions on Ω. By Montel’s theorem, there is a subsequence ψjk that converges uniformly on compact subsets of Ω to some element ψ0 ∈ O(Ω). We assert that ψ00 (ϕ0(w)) · ϕ00 (w) ≡ 1 for all w ∈ Ω with ϕ0(w) ∈ Ω .

(3.1.1)

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To see the assertion, observe that ψj ◦ϕj = id and hence ψj0 (ϕj (z))·ϕ0j (z) = 1 for all j and all z ∈ Ω. Thus we need only check that lim ψj0 (ϕj (w)) = ψ00 (ϕ0(w))

j→∞

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for all w ∈ Ω∩ϕ−1 0 (Ω). This assertion holds, however, because limj→∞ ϕj (w) = ϕ0 (w) and the sequence ψj0 converges uniformly on compact sets to ψ00 . With our claim proved, we now note that, if ϕ0 is not constant, then ϕ0 is certainly a holomorphic map from Ω to Ω. Also ψ0 is not constant by (3.1.1), and again we see that ψ0 is a holomorphic map from Ω to Ω. Since ψj ◦ ϕj = id = ϕj ◦ ψj , we conclude that ψ0 ◦ ϕ0 = id = ϕ0 ◦ ψ0 , hence ϕ0 ∈ Aut (Ω). But if ϕ0 is constant then (3.1.1) tells us that c ≡ ϕ0 (Ω) cannot be a point in Ω. It follows that c ∈ ∂Ω.

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In what follows, we use D to denote the unit disc in the complex plane.

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Theorem 3.2 Let Ω be a bounded domain in C with C 1 boundary. If Ω has noncompact automorphism group, then Ω is conformally equivalent to the unit disc D.

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Remark 3.3 It will be clear from the proof that C 1 boundary is not the sharp condition here. We do not know the precise condition for the optimal formulation of this theorem.

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Proof of the Theorem: By the Riemann mapping theorem, it suffices to prove that Ω is simply connected. We establish this fact in several steps. We claim that there must be a point P ∈ Ω and elements ϕj ∈ Aut (Ω) such that ϕj (P ) accumulates at a boundary point X ∈ ∂Ω. If this were not the case, then for any collection {ψj } ⊆ Aut (Ω) and any fixed P ∈ Ω it would hold that {ψj (P )} lies in a fixed, compact subset L ⊆ Ω. Of course Aut (Ω) is a normal family (since Ω is bounded), hence there is a normally convergent subsequence ψjk that converges to some limit holomorphic function ψ0. Now Hurwitz’s theorem (or the argument principle) shows easily that this limit function ψ0 is either univalent or constant (and, incidentally, the constant case can be eliminated using Theorem 3.1 above). And the fact that all the ψjk (P ) are trapped in L shows that

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(1) There is a noncompact orbit.

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the limit function maps Ω into Ω. Since similar reasoning applies to the inverse mappings, we may conclude that the limit mapping ψ0 is in fact an automorphism. So we would see that Aut (Ω) is compact. Since we assumed that in fact this is not the case, we may therefore conclude that the asserted points P ∈ Ω and X ∈ ∂Ω and automorphisms ϕj must exist. A “peaking function” at the point X is a function µ : Ω → D such that

(2) There is a peaking function at the point X ∈ ∂Ω.

• µ is continuous on Ω;

• µ is holomorphic on Ω;

• µ(X) = 1;

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• |µ(ζ)| < 1 for all ζ ∈ Ω \ {X}.

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Such a peaking function may be constructed by mapping Ω univalently, with a mapping ξ, into the unit disc so that the holes in Ω go to interior regions in the disc and the outer boundary of Ω goes to the boundary of the disc. Carath´eodory’s theorem (see [GRK8] or any standard complex analysis text) tells us that the mapping extends continuously and univalently to the boundary. See Figure 1. Suppose that the boundary point X ∈ ∂Ω gets mapped to 1 ∈ ∂D. Now the function f(ζ) =

ζ +1 2

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is a peaking function for the point 1 in ∂D. Thus µ ≡ f ◦ ξ is the peaking function that we seek for Ω at X. (3) If K ⊆ Ω is any compact subset and if ψj and P, X are as in asser-

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tion (1) then there is a subsequence ψjk such that ψjk (z) → X uniformly over z ∈ K. To see this, let ψj , P, X be as in part (1). Let µ be the peaking function as in part (2). Consider the maps λj ≡ µ◦ψj . Observe that each λj is holomorphic and bounded by 1. By Montel’s theorem, there is a subsequence λjk that converges uniformly on compact sets to a limit function λ0 . Of course |λ0 (ζ)| ≤ 1 for all ζ ∈ Ω. But observe that

λ0 (P ) = lim λj (P ) = lim µ(ψj (P )) = µ( lim ψj (P )) = µ(X) = 1 . j→∞

j→∞

j→∞

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Figure 1: Continuous and univalent extension of the mapping.

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We have contradicted the maximum modulus principle (see [GRK1] or any standard complex analysis text) unless λ0 ≡ 1. But then this means that the ψjk are converging, uniformly on compact sets, to the constant function with value X. That is what has been claimed. (4) There is a small, open disc D centered at X such that Ω∩D is sim-

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ply connected. In fact this is a straightforward application of the implicit function theorem (see [KRP1] for a comprehensive treatment, or [RUD] for a more basic introduction), and we leave the details to the reader. Suppose not. Then there is a closed loop γ : [0, 1] → Ω that cannot be continuously deformed in Ω to a point. But of course the image curve of γ is a compact set C. Let D be the disc that we found in step (4). By step (3), there is a k so large that ψjk (C) ⊆ D ∩ Ω. But this means that ψjk ◦ γ is a closed curve that lies entirely in the simply connected region D ∩ Ω. Hence ψjk ◦ γ can certainly be deformed to a point inside D ∩ Ω. But of course the map ψj is a homeomorphism. Hence the image C of γ itself may deformed to a point in Ω. That is a contradiction.

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(5) The domain Ω is simply connected.

(6) And now our proof is complete, for the simply connected domain Ω is

(by the Riemann mapping theorem) conformally equivalent to the disc. That is what was claimed. 7

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There is a similar result in several complex variables, due to Bun Wong [WON1] and Rosay [ROS]. We shall consider it in the next section. We shall finish this section with some representative results about automorphisms on domains in complex dimension one. We shall prove some of these results, but only sketch proofs for others. Proposition 3.4 Let Ω ⊆ C be a bounded domain. Let P ∈ Ω and suppose that φ : Ω → Ω satisfies φ(P ) = P. If φ0 (P ) = 1, then φ is the identity.

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Proof: We may assume that P = 0. Expanding φ in a power series about P = 0, we have φ(z) = z + Pk (z) + O(|z|k+1 ),

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where Pk is the first nonvanishing monomial (of degree k) of order exceeding 1 in the Taylor expansion. Notice that   1 ∂k φ(0) z k . Pk (z) = k! ∂z k Defining φj (z) = φ ◦ · · · ◦ φ (j times), with φ0 = id, we have

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φ2(z) = z + 2Pk (z) + O(|z|k+1 ) , φ3(z) = z + 3Pk (z) + O(|z|k+1 ) , .. . j φ (z) = z + jPk (z) + O(|z|k+1 ).

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Choose discs D(0, a) ⊆ Ω ⊆ D(0, b). Then, for 0 ≤ j ∈ Z, we know that D(0, a) ⊆ dom φj ⊆ D(0, b). Therefore the Cauchy estimates imply that, for our index k, we have k k ∂ ∂ j b · k! j k φ(0) = k φ (0) ≤ k . ∂z ∂z a

Letting j → ∞ yields that ∂ k φ/∂z k (0) = 0. We conclude that Pk = 0; this equality contradicts the choice of Pk unless we have φ(z) ≡ z. 8

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In this last result, if one equips the domain with an invariant metric such as the Bergman metric, then one can interpret the theorem as a standard uniqueness result for isometries. We know that the dimension of the automorphism group of the disc is three (since the automorphism group consists of rotations and M¨obius transformations). And the dimension of the automorphism group of the annulus is one (since the automorphism group consists of rotations and the reflection). The following is a classical result of M. Heins (see [HEI1], [HEI2]). We shall use the standard terminology that a domain has connectivity k ≥ 1 if the complement of the domain has k connected components.

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Theorem 3.5 Let Ω ⊆ C be a domain with finite connectivity at least three (i.e., the domain has at least two holes). Then the automorphism group is finite.

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Next we turn to the study of iterates of one automorphism. In what follows, if f : Ω → Ω is a holomorphic mapping, then we shall use the notation f j (z) = f ◦ f ◦ · · · ◦ f for j = 1, 2, . . . . | {z }

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Proposition 3.6 Let f be a holomorphic mapping of the domain Ω to itself. Assume that some subsequence f jk converges normally to a function g ∈ O(Ω). We conclude that

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(a) If g ∈ Aut (Ω), then f ∈ Aut (Ω).

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(b) If g is not constant, then every convergent subsequence of the sequence hk ≡ f jk+1 −jk has the limit function idΩ .

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Proof: First consider part (a). If it were the case that f(a) = f(b), some a, b ∈ Ω, then it would follow that f j (a) = f j (b) for all j and hence g(a) = g(b). We must conclude that a = b since g ∈ Aut (Ω). For surjectivity, we know that f jk (Ω) ⊆ f(Ω) for every choice of index. It is immediate that g(Ω) ⊆ f(Ω) ⊆ Ω, just by set theory. But we must have that g(Ω) = Ω because g is an automorphism. Therefore f(Ω) = Ω. Assertion (a) is now proved. We turn to (b). We certainly know that g is a holomorphic mapping from Ω to Ω. Let h be some subsequential limit of the hk . Then fjk+1 = hk ◦ fjk 9

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implies that g = h ◦ g. Therefore h is the identity on g(Ω). But g(Ω) is open in Ω, so we know that h ≡ idΩ . That completes the proof. One interesting consequence of this last proposition is that, if the iterates f of f converge uniformly on compact sets to a nonconstant function, then the limit function is the identity. Now we have an important result of H. Cartan: j

Theorem 3.7 ([H. Cartan]) Let Ω be a bounded domain and let f be a holomorphic mapping of Ω to Ω. Suppose some subsequence f jk of the iterates of f converges uniformly on compact sets to a nonconstant function g. Then f ∈ Aut (Ω).

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Proof: Montel’s theorem tells us that hk ≡ f jk+1 −jk has a subsequence that converges uniformly on compact sets. By part (b) of the last proposition, we know that the limit of this subsequence is idΩ . By part (a) of that proposition, f ∈ Aut (Ω).

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The next result was proved only in the last forty years. See [FIF], [MAS]. The proof that we present is a more modern one that comes from [KRA6]. The reader should note that no hypothesis is made about the topology of the domain in question.

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Theorem 3.8 Let Ω ⊆ C be any bounded, planar domain and φ : Ω → Ω an automorphism. If φ has three fixed points (i.e., φ(p) = p for three distinct points p), then φ is the identity mapping.

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Proof: We must begin with some terminology. We take it for granted that the reader has at least an intuitive understanding of the concept of “geodesic” (see [STO] or [BOO] for background). Going by the book, a geodesic is defined by a differential equation. For our purposes here, we may think of a geodesic as a locally length-minimizing curve. Let x ∈ M, where M is a Riemannian manifold. A point y ∈ M is called a cut point of x if there are two or more length-minimizing geodesics from x to y in M. See Figure 2. We further use the following basic terminology and facts from Riemannian geometry. Let dis(x, y) denote the metric distance from x to y. A geodesic γ : [a, b] 7→ M is called a length-minimizing geodesic (or alternatively, a minimal geodesic, or a minimal connector) from x to y if γ(a) = 10

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Figure 2: The point y is a cut point for x.

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x, γ(b) = y, and dis (x, y) = [arc length of γ]. By the Hopf-Rinow theorem, any two points in a complete Riemannian manifold can be connected by a minimizing geodesic. If there is a smooth family of minimizing geodesics from x to y, then these two points are said to be conjugate. Conjugate points are cut points. The collection of cut points of x in M is called the cut locus of x, which we denote by Cx. It is known that Cx is nowhere dense in M ([GKM], [KLI], e.g.), in fact, Cx lies in the singular set for the distance function. Now equip Ω with the Bergman metric. For convenience, we shall suppose that Ω has C 1 boundary. This will guarantee that the Bergman metric is complete (see [OHS]). We assume that f ∈ Aut (Ω) is not the identity map, but has 3 distinct fixed points a, b, c in Ω. To reach a contradiction, let us start with the fixed point a. If the set of fixed points accumulates at a, we are done. So we may choose as the second fixed point b the closest (with respect to the Hermitian metric) one to a apart from a itself. This choice may not be unique, and hence we simply choose one. We need to consider only the case that b is a cut point (not conjugate) of a. [For otherwise f would fix the unique geodesic connecting a to b and hence would be the identity.] Then there will be several unit-speed minimal connectors (all of which have the same length, of course), say γ1 , γ2 , . . ., joining a to b. First notice that no minimal connector can have a self-intersection. Then the automorphism f maps any one of the minimal connectors to an11

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other such, since the endpoints a and b are fixed. Note that f ◦ γ1 cannot intersect γ1 except at the endpoints. For, if they do intersect at a point other than the endpoints, then they have to intersect at the same time; otherwise one may find an even shorter connector between a and b than the minimal connector, which is a contradiction. Then the intersection point becomes a fixed point of f closer to a than b, which is again not allowed. Now, γ1 and f ◦ γ1 join to form a piecewise smooth Jordan curve in the plane. Thus it bounds a cell, say E, in the plane C. Now consider the third fixed point c that is distinct from a and b. Notice that we may assume that c is not on any of the minimal connectors for a and b. Suppose that c is inside the cell E. Now join c to a by an arc ξ in E ∩ Ω that does not intersect with either γ1 or f ◦ γ1 , or in fact with any minimal geodesics joining a and b. Notice that the conformality of f at the fixed point a shows that there is a sufficiently small open disc neighborhood U of a on which f must map U ∩ ξ to the outside of the cell E. This results in the conclusion that f ◦ ξ must cross γ1 or f ◦ γ1 . But this is impossible, since a point not on any minimal connector from a to b cannot be mapped to a point on a minimal connector from a to b. If c is outside the cell E, then the arguments are similar. Since there are only finitely many minimal connectors between a and b (since a and b are not conjugate to each other, and the quotient from the universal covering space is formed by a properly discontinuous group action; see [BRE]), some iterate f m of f will move ξ so that its image has points inside E. Then, f m ◦ ξ again crosses one of these minimal geodesics joining a and b, which leads us to another contradiction.

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The next result is philosophically related to the three-fixed-point theorem (Theorem 3.11) that we just proved. Corollary 3.9 Let Ω be a bounded domain in the plane and suppose that f : Ω → Ω is a holomorphic mapping with two distinct fixed points. Then f is an automorphism of Ω.

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Proof: Call the fixed points a and b with a 6= b. We assume, of course, that a 6= b. Montel’s theorem tells us that there is a subsequence {f jk } of the iterates that converges in Ω to a holomorphic function g. Certainly g(a) = a and g(b) = b. Hence g is not constant. By Theorem 3.10, f ∈ Aut (Ω). 12

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The next corollary is in part a reiteration Proposition 3.4. Compare it to the classical Schwarz lemma. Corollary 3.10 Let Ω be a bounded domain and a ∈ Ω. Let Aut a (Ω) denote the collection of automorphisms of Ω that fix the point a. We call this the isotropy subgroup of a. Then we have Aut a (Ω) = {f : f is a holomorphic mapping from Ω to Ω , f(a) = a , |f 0 (a)| = 1} .

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Proof: As usual, by Montel, there is a subsequence f jk of iterates that converges, uniformly on compact sets, to some holomorphic g. One may calculate that lim [f jk ]0(a) = lim [f 0 (a)]jk = g 0 (a) .

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But this can be true only if |f (a)| ≤ 1 (otherwise the powers of the derivative would blow up). In case |f 0 (a)| = 1, we see that |g 0 (a)| = 1. But then g is not constant, and H. Cartan’s theorem tells us that f ∈ Aut (Ω). Conversely, suppose that f ∈ Aut (Ω) and f fixes a. Then f −1 ∈ Aut a (Ω). We know already that |f 0(a)| ≤ 1. Likewise, |1/f 0 (a)| = |(f −1 )0 (a)| ≤ 1, or |f 0 (a)| ≥ 1. We conclude that |f 0 (a)| = 1.

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Proposition 3.11 Let Ω be a bounded domain. Fix a point a ∈ Ω. Let f ∈ Aut a (Ω). If f 0 (a) > 0, then f = idΩ .

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Proof: We know that f 0 (a) lies on the circle. If f 0 (a) > 0, then f 0 (a) = 1. It follows now from H. Cartan’s theorem that f = idΩ .

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Theorem 3.12 Let Ω be a bounded, finitely connected domain in C with connectivity at least 2. Assume that each connected component of the complement of Ω has at least two points. Let f be a holomorphic self-map of Ω. Assume that every closed path γ in Ω that is not homologous to 0 in Ω has image under f that is also not homologous to 0 in Ω. Then f ∈ Aut (Ω). 13

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Proof: As usual, we apply Montel’s theorem to the sequence of iterates f j . We find thereby a subsequence f jk that converges to some limit g. Because Ω has connectivity at least 2 (i.e., the complement has at least two connected components), there is a closed path γ in Ω that is not homologous to 0. Since f j ◦ γ = f ◦ (f j−1 ◦ γ), we see that no path f jk ◦ γ can be homologous to 0 in Ω. Thus, for each k, there is a connected component Kk of the complement of Ω such that Kk lies in the interior region of the closed curve f jk ◦ γ. Since Ω has just finitely many holes (i.e., bounded connected components of the complement), we may suppose (by passing to a subsequence if necessary and renumbering the holes) that Kk lies in the region interior to f jk ◦ γ for each k. Since Kk has at least two points, we find from our preceding proposition that g is not constant. Now Proposition 3.9 tells us that f is an automorphism of Ω.

The Several Complex Variables Context

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There is no Riemann mapping theorem in several complex variables. Indeed, Poincar´e proved in 1906 that the unit ball and the unit bidisc in C2 are biholomorphically distinct. We now know, thanks to work of Burns/Shnider/Wells [BSW] and Greene-Krantz [GRK1], that two smoothly bounded domains in Cn , n > 1, are generically biholomorphically distinct. Indeed, it is important to understand here the geometric distinction between one complex dimension and several complex dimensions. For a domain in one complex variable, the tangent space to a boundary point is a real line. It has no complex structure. So there cannot be any complex biholomorphic invariants. That is why it is possible for there to be a Riemann mapping theorem in dimension one. In Cn , n > 1, the tangent space to a boundary point of a domain has real dimension 2n − 1. And it will contain a complex subspace of complex dimension n − 1. It is definitely the case that complex biholomorphic invariants live in that subspace. The seminal paper [CHM] calculates these invariants when the domain in question is strongly pseudoconvex or has nondegenerate Levi form. Further developments along those lines appear in [FEF2] and [GRA]. See also [GON] and [MOW]. Indeed, the literature on normal forms is considerable. We cannot begin to desribe it here, but only mention that some of the key workers in the subject area are S.-S. Chern, V. Ezhov, V. Guillemin, X. Huang, A. Isaev, J. Lebl, J. Moser, 14

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A. Tromba, S. Webster, and D. Zaitsev, The situation in several complex variables is complicated. Chern/Moser invariants are very difficult to understand, and there is no comparable set of invariants (that have been explicitly calculated) for weakly pseudoconvex domains. So we seek other biholomorphic invariants that will help us to understand biholomorphic equivalence classes of domains. Sometimes useful here is the theory of automorphism groups, although we must note that Chern-Moser invariants always exist while automorphism groups can be trivial. It is clear that, if a domain Ω1 is biholomorphic to a domain Ω2 via a mapping Φ, then their respective automorphism groups are isomorphic via the mapping

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Aut (Ω1 ) 3 ϕ 7−→ Φ ◦ ϕ ◦ Φ−1 ∈ Aut (Ω2 ) .

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This mapping is a group isomorphism. In fact Poincar´e’s original proof that the ball and the bidisc are biholomorphically inequivalent consisted in showing that the automorphism groups cannot be isomorphic. There are today a number of other ways (some of them geometric, some analytic) to show that these domains are inequivalent. The converse to the statement in the last paragraph is of course false. Let Ω1 = {ζ ∈ C : 1 < |ζ| < 2} and Ω2 = {ζ ∈ C : 1 < |ζ| < 4}. Then Ω1 and Ω1 have the same automorphism group (rotations plus inversion), yet they are certainly not conformally equivalent. An example in higher dimensions may be obtained by taking the product of each of these domains with a ball. We shall begin the next section by discussing the theorem of Bun Wong which has been the wellspring of much of the activity surrounding automorphism groups of domains in several complex variables.

The Theorem of Bun Wong and Rosay

The original theorem of Bun Wong [WON1] says this:

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Theorem 5.1 Let Ω ⊆ Cn be smoothly bounded and strongly pseudoconvex. Assume that the automorphism group of Ω is noncompact. Then Ω is biholomorphic to the unit ball B in Cn . A few years later, Rosay [ROS] generalized the result as follows: 15

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Theorem 5.2 Let Ω ⊆ Cn be any bounded domain. Let X ∈ ∂Ω and assume that there is a boundary neighborhood U of X so that each point of ∂Ω ∩ U is strongly pseudoconvex. Further suppose that there are a point P ∈ Ω and automorphisms ϕj ∈ Aut (Ω) so that ϕj (P ) → X as j → ∞. Then Ω is biholomorphic to the unit ball B ⊆ Cn .

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Rosay’s theorem inspired a good deal of additional work, for it suggested that the Levi geometry at or near a boundary orbit accumulation point (i.e., the point that we call X) determines the global geometry of the domain. We call X a boundary orbit accumulation point. The first paper to explore this new point of view was [GRK4]. Wong’s original proof of his theorem consisted in constructing a special metric on Ω. Rosay’s proof was pure function theory (see [KRA1] for all the details). The paper [KIK1] gives a proof using techniques from function algebras. The proof using the scaling method (also known in differential geometry circles as the technique of flattening) is described in detail in Section 10 below. See also [GKK]. The original proof of the Bun Wong/Rosay theorem is rather complicated, and we cannot describe it here. Suffice it to say that the basic idea is this: • Let K be a large compact subset of Ω. • Since X is strongly pseudoconvex, it is a peak point. It follows that X is a point of completeness for the Bergman metric on Ω.

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• Let U be a small neighborhood of X. It follows from the above that, if j is large, then ϕj (K) ⊆ U.

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• By analysis of the Levi form, one can see that ∂Ω near X is asymptotically like a ball. • It then follows that U ∩ Ω can be mapped nicely into a boundary neighborhood of the ball B. Thus K is mapped in as well.

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• Now, by applying an automorphism of the ball, the image of K described in the last step may be mapped to a large compact set L in B.

• Summarizing, we have a biholomorphic map from a large compact set K in Ω to a large compact set L in B. Exhausing Ω by such compact 16

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sets (and, correspondingly, exhausting B by image compact sets), we get in the limit a biholomorphism from Ω to B. The study of issues connected to the Bun Wong/Rosay theorem led to the Greene/Krantz conjecture, which is the topic of the next section.

6

The Greene/Krantz Conjecture

Let us begin by recalling the notion of finite type (originally due to Kohn, see [KOH]). We begin in dimension 2. Let Ω = {z ∈ C2 : ρ(z) < 0}. Assume that we are working near a boundary point at which ∂ρ/∂z2 6= 0. Then the vector field

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∂ρ ∂ ∂ρ ∂ − ∂z1 ∂z2 ∂z2 ∂z1

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T =

is tangential (because it annihilates ρ). We let = spanC {T, T } = spanC {L0 , [T, T ]} = spanC {L1 , [T, S], [T, S] : S ∈ L1 } etc.

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L0 L1 L2

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We say that a point P ∈ ∂Ω is of finite type m if Lm−1 contains no vector field with a complex normal component at P , but Lm does contain a vector field with a complex normal component at P . Put in other words, hY, ∂ρi = 0 for all Y ∈ Lm−1 and hZ, ∂ρi = 6 0 for some Z ∈ Lm . What we have just described is the analytic definition of finite type. There is also a geometric definition of the concept. It is formulated as follows. We say that a one-dimensional complex analytic curve ψ(ζ) with ψ(0) = P has order of contact k with ∂Ω at the boundary point P if ρ ◦ ψ vanishes to order k at 0. Now P is said to be of geometric finite type m if it has a one-dimensional complex analytic curve with order of contact m at P , but it does not have a one-dimensional complex analytic curve with order of contact m + 1 at P . This notion of geometric type was implicit in Kohn’s original paper [KOH] and was made quite explicit in the paper [BLG]. It was 17

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D’Angelo [JPDA1] who found a way to formulate the concept of geometric finite type in higher dimensions. Now the important result about these ideas is as follows: Theorem 6.1 Let Ω ⊆ C2 be smoothly bounded and let P ∈ ∂Ω. Then P is of analytic finite type m if and only if P is of geometric finite type m.

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See [KOH] and [BLG] for the genesis of this result. We cannot prove the theorem here, but refer the reader to [KRA1, §11.5] for all the details. Interestingly, the situation for finite type in dimensions 3 and higher is not so simple. The reason is that holomorphic varieties in C2 are 1-dimensional, and therefore have no interesting subvarieties. But holomorphic varieties in Cn for n ≥ 3 do have interesting subvarieties, and that makes life complicated. It was not until work of D’Angelo (see [DAN1], [DAN2]) that the idea of finite type in higher dimensions was fully understood. It is a complicated story, and we cannot treat it in any detail here. Nontrivial ideas from algebraic geometry are involved. We refer the reader to [KRA1, §11.5] for a leisurely treatment of the subject. See also [JPDA2]. Suffice it to say that the purpose of the concept of finite type is to find a way to measure lack of complex structure in the boundary of a domain. Now we can formulate the Greene/Krantz conjecture:

CONJECTURE (Greene/Krantz): Let Ω ⊆ Cn be a smoothly

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bounded, pseudoconvex domain. Let X ∈ ∂Ω be a boundary orbit accumulation point. Then X must be of finite type.

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There is considerable ad hoc evidence for the Greene/Krantz conjecture. Bedford and Pinchuk (see [BEP1]) proved that, if Ω ⊆ C2 is a bounded domain with real analytic boundary and noncompact automorphism group (hence P and X exist), then Ω must have the form Em ≡ {(z1, z2) ∈ C2 : |z1|2 + |z2|2m < 1}

for some positive integer m. Catlin (unpublished) observed that their hypothesis could be weakened to finite type in the sense of Kohn. In the paper [BEP2], Bedford and Pinchuk go on to prove that, if Ω ⊆ Cn+1 is a smoothly bounded, pseudoconvex domain of finite type and such that the Levi form 18

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has rank at least n−1 at each boundary point, and if Aut (Ω) is noncompact, then Ω is biholomorphically equivalent to a domain of the form ( ) n X Em ≡ (w, z1 , z2, . . . , zn ) ∈ Cn+1 : |w|2m + |zj |2 < 1 . j=1

In [BEP3], these authors treat smoothly bounded, finite type, convex domains in Cn+1 . Assume as usual that the domain has noncompact automorphism group. Then Ω is biholomorphically equivalent to a domain of the form  E = (w, z1 , z2, . . . , zn ) : |w|2 + p(z, z) < 1 ,

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where p is a weighted, homogeneous, real polynomial. Finally, in [BEP4], Bedford and Pinchuk remove the hypothesis of “pseudoconvex” from the theorem in [BEP1]. All four of these papers involve delicate scaling arguments, sometimes applied multiple times. See Section 10 for a discursive discussion of the scaling method. Kim and Krantz [KIK3] proved a version of the Greene/Krantz conjecture for a restricted class of domains. Scaling (see [PIN] and [FRA] for the genesis of this idea and [GKK] for a leisurely exposition) was a principal tool in that proof. Kim [KIM2] proved that, if the boundary orbit accumulation point is contained in an analytic disc in the boundary, then the domain must be a product. That is certainly a contrapositive version of the conjecture. A version of the Greene/Krantz conjecture was proved by Kim [KIM3] and Wong [WON2] in the convex case and by Fu and Wong [FUW1], [FUW2] in the general pseudoconvex case. It says this. Let Ω ⊆ C2 be a bounded domain with C 4 boundary, and let P ∈ ∂Ω be a boundary orbit accumulation point. Then ∂Ω cannot contain a nontrivial germ of a complex variety at P . This does not say that ∂Ω is of finite type at P (the domain Ω = {(z1 , z2) ∈ C2 : |z1|2 + 2exp(−1/|z2 |2) < 1} meets the condition at (1, 0) but is not of finite type at that point), but it points in that direction. See also [KRA3] for recent results. Other versions of the Greene/Krantz conjecture have been confirmed by Verma [VER] and others. We are still nowhere close to proving the full conjecture, but it is widely believed to be true.

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7

The Concept of Semicontinuity

The idea of semicontinuity of automorphism groups was first introduced in [GRK5] and developed further in [GRK6], [GKKS], and [KRA4]. In order to formulate the result we need to define a new topology on domains. Fix a smoothly bounded domain Ω0 in Cn . Say that Ω0 = {z ∈ Cn : ρ0 (z) < 0}. For  > 0, define a subbasis element of the C ∞ topology on domains centered at Ω0 to be   UΩ0 , = Ω = {z : ρ(z) < 0} : kρ − ρ0 kC ∞ <  .

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We see that UΩ0 , is a collection of domains. We call the topology generated by this subbasis the C ∞ topology on domains. Now we have the following result.

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Theorem 7.1 (Greene/Krantz) Let Ω0 ⊆ Cn be a smoothly bounded, strongly pseudoconvex domain. Then there is a neighborhood U of Ω0 in the C ∞ topology on domains so that, if Ω ∈ U, then Aut (Ω) is a subgroup of Aut (Ω0 ). Furthermore, there is a C ∞ mapping Ψ : Ω → Ω0 so that Aut (Ω) 3 ϕ 7−→ Ψ ◦ ϕ ◦ Ψ−1 ∈ Aut (Ω0 ) is an injective homomorphism of groups.

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It is particularly noteworthy that we cannot suppose that the mapping Ψ in the theorem is a biholomorphism. As we have already observed, the domains Ω and Ω0 will be generically not biholomorphic. The theorem makes concrete and explicit the intuitive notion that small perturbations cannot create symmetry, but they can certainly destroy symmetry. A sketch of the proof of Theorem 7.1 is as follows:

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• One first shows that, under small, smooth perturbations of the boundary of Ω0 , estimates for the ∂-Neumann problem only vary by a small amount. • Assuming that Ω0 is not the ball (there is a separate, much easier argument in the ball case), we know by Bun Wong’s theorem that the automorphism group is compact.

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• Combining the first two facts, we can use an averaging procedure to produce a smooth, biholomorphically invariant metric on Ω0 . • With a little geometric analysis, we can arrange for the metric in the last step to have a product structure near the boundary. • Now we can form the metric double of Ω0 to obtain a compact Riemanb 0 equipped with an invariant metric. nian manifold Ω

• Finally, a classical result of Ebin [EBI] applies to give a semicontinuity b 0. result for isometries of Ω

• Some careful analysis then allows us to derive a semicontinuity result for automorphisms of Ω0 .

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So we see that the automorphism group of a small perturbation of a fixed, base domain will be generically “smaller” than the automorphism group of the base domain. Of course it should be noted that one cannot reverse the roles of Ω and Ω0 in the statement of the theorem, thereby concluding that Aut (Ω0 ) is a subgroup of Aut (Ω) (and therefore that the two groups are isomorphic). This assertion is false. The point is that the size of the allowed perturbation depends on the geometry of the base domain. In the paper [KRA4], this author generalizes this semicontinuity theorem to a class of finite type domains in C2 . His argument follows the broad strokes of [GRK5], but for one important point. An essential feature of the argument in [GRK5] is that it is shown that derivatives of automorphisms on a strongly pseudoconvex domain with compact automorphism group are uniformly bounded—uniformly bounded over the closure of the domain and uniformly bounded over all the elements of the group. This information is used in turn to construct an invariant metric which is smooth on the closure of the domain, and which is a product metric near the boundary. With such a metric, one can form the metric double and then use tools of analysis on compact manifolds. In the paper [GRK5], these uniform estimates on automorphism derivatives are obtained using Bergman representative coordinates, and that calculation depends in an essential way on Fefferman’s asymptotic expansion in [FEF1]. These devices are not available on a domain of finite type in C2 . So [KRA4] needed to develop other tools in order to follow the same path. In the paper [GKKS], we introduce new arguments to produce a semicontinuity theorem on strongly pseudoconvex domains with minimal boundary 21

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smoothness (just C 2). The rather delicate proof uses the scaling method of Pinchuk [PIN], Frankel [FRA], and others—see Section 10 below. K.-T. Kim [KIK2] has made contributions to this study. The result in [GKKS] may be compared with a similar theorem in the paper [GRK6].

8

Characterization of Domains by their Automorphism Groups

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The automorphism group of a bounded domain, when topologized by uniform convergence on compact sets (i.e., the compact/open topology) is a real Lie group. In particular, it is finite dimensional—just because a Lie group is by definition locally Euclidean. The automorphism group is never a complex Lie group (see [KOB1]) unless the dimension of the group is zero (which is in fact generically the case). A standard heuristic is that, among bounded domains, the unit ball has the “largest” automorphism group. Among unbounded domains in Cn , we perceive that Cn itself has the “largest” automorphism group (indeed—see below—it is infinite dimensional). In the present subsection we formulate these results analytically and discuss some of the proofs. Related results are also presented. Of course it is well known that the automorphism group of the unit ball in Cn is SU(n, 1)/C, where C is the center. This group has dimension n2 + 2n. The automorphism group of Cn is quite large. It is certainly not a Lie group. In particular, Aut (C2 ) contains the shears (z, w) 7−→ (z, w + f(z)), for any entire function f of one complex variable. So Aut (Cn ) is plainly infinite dimensional. It is known that, in a certain sense, the shears generate Aut (Cn ) (see [ANL], [ROR]). Nevertheless, the automorphism group of Cn is quite unwieldly, and the standard tools for studying automorphism groups are unavailable in that context. In particular, normal families arguments do not apply. The characterization of the unit ball is also treated in the paper [ISSK2]. The characterization of Cn is treated in greater generality in the seminal work [ISK1] of Isaev and Kruzhilin.

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8.1

Enunciation of Basic Results

n In z n ) ∈ Cn : P what2 follows, B ⊆ C denotes the unitn ball B ≡ {z = (z1, . . . ,∼ j |zj | < 1}. If Ω1 , Ω2 are domains in C then we will write Ω1 = Ω2 if Ω1 is biholomorphic to Ω2 . Our first result can be stated, in its essence, as follows (see [KAUP] and [KOB1]):

Theorem 8.1 Let Ω ⊆ Cn be a bounded domain. If Aut(Ω) has the same dimension as the dimension of Aut(B) then Ω ∼ = B. A simplified, but incisive, formulation of our second result is this (for which see [ISK1]):

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Theorem 8.2 Let Ω ⊆ Cn be pseudoconvex. Assume that Aut(Ω) and Aut(Cn ) are isomorphic as topological groups. Then Ω ∼ = Cn . EXAMPLE 8.3 Let m be an integer, m ≥ 2. Define

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Ωm ≡ {(z1, z2) ∈ C2 : |z1|2m + |z2|2m < 1}.

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Then, regardless of the choice of m, the automorphism group of Ωm is generated by S 1 × S 1 and the operation of switching the two variables (see [GKK] for a discussion of this and related matters). And the Ωm are of course biholomorphically distinct for different values of m (for which see [BEL], [WEB1]). So these domains cannot be characterized by their automorphism groups.

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EXAMPLE 8.4 Let Ω = C2\{0}. Then any mapping of the form (z, w) 7−→ (z, w + g(z)), where g is an entire function of one complex variable that vanishes at the origin, is an automorphism of Ω. Of course the space of all such g is infinite dimensional. Thus Aut(Ω) is infinite dimensional (not a Lie group). Certainly the group is not locally compact. The automorphism group of C2 is also infinite dimensional (not a Lie group). [Of course the Hartogs extension phenomenon tells us that any automorphism of the above Ω automatically analytically continues to an automorphism of C2 .] Yet these two domains are not biholomorphic—they are not even homeomorphic.

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EXAMPLE 8.5 Let G be any compact Lie group. The paper [BED] (see also [SAZ]) shows that there is a smoothly bounded, strongly pseudoconvex domain Ω whose automorphism group is precisely G. In fact Ω may be taken to have real analytic boundary and be arbitrarily close to the unit ball in some Cn (see [GKK]). Yet the construction in [BED] reveals many choices in the construction of Ω. There will be uncountably many biholomorphically distinct domains with automorphism group G. [To see this last claim, only note that, for any given natural number `, there are uncountably many distinct domains with automorphism group T` . Also the proof of Proposition 3.1 in [BED] shows that there are uncountably many choices for each of the domains Ωj . Our result follows.]

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It is a general principle—and the last example illustrates it concretely— that theorems like 8.1 and 8.2 are only possible when the automorphism group in question is non-compact.

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EXAMPLE 8.6 Let ϕ ∈ Cc∞ (C2 ) be non-zero, real-valued, have support in a ball of radius 1/100 centered at the origin, be identically equal to 1/100 on a ball of radius 1/200 centered at the origin, and satisfy |ϕ(z)| ≤ 1/100 for all z. Set Ω0 = {(z1, z2) ∈ C2 : |z1|2 + |z2|2 + ϕ(z1, z2 − 1) < 1} . Now define

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Finally, let

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Φ(z1 , z2) =

! p 3/4z2 z1 − 1/2 , . 1 − [1/2]z1 1 − [1/2]z1

Ω=

∞ [

Φj (Ω0 ) .

j=−∞

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[Here Φ raised to the j th power, for j > 0, denotes Φ ◦ · · · ◦ Φ taken j times; also Φ raised to the j th power, for j < 0, denotes Φ−1 ◦ · · · ◦ Φ−1 taken |j| times; finally Φ0 = id.] Then it is not difficult to verify (see [LER]) that Aut(Ω) = Z. Moreover, different choices of ϕ will generically produce biholomorphically distinct domains from this construction (alternatively, one could choose different M¨obius transformations Φ to produce biholomorphically distinct domains). So the domains Ω cannot be characterized by their 24

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automorphism groups, even though Aut(Ω) is non-compact.

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EXAMPLE 8.7 Consider domains in the complex plane C. Any finitely connected domain of connectivity at least three has automorphism group which is a finite group—see [HEI2]. It is easy to see—using extremal length, for example—that there will be infinitely many distinct domains with any given finite automorphism group.6 The only planar domain with a one-dimensional automorphism group is the annulus (the group is two copies of the circle). The only bounded planar domain with automorphism group having dimension (at least) two is the disc—and in fact the automorphism group of the disc has dimension three (since the M¨obius transformations are parametrized by two real parameters and the rotations by one). For unbounded domains, the punctured plane has automorphism group of dimension two (rotations and dilations and inversion) and the entire plane has automorphism group of dimension four (all maps of the form z 7−→ az +b with a, b complex constants).

Discussion of the Proof of Theorem 8.1 and Its Generalizations

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In fact it is convenient to prove this more general version of Theorem 8.1 (see [ISSK2]):

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Theorem 8.8 Let M be a connected Kobayashi-hyperbolic manifold of complex dimension n ≥ 2. Suppose that dim Aut(M) ≥ n2 + 3. Then M is biholomorphically equivalent to the unit ball B ⊆ Cn .

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Remark 8.9 The papers [KAUP], [KOB1] prove a result like this one with n2 + 3 replaced by n2 + 2n (the dimension of the automorphism group of the ball). 6

On the other hand, it is not known precisely which groups arise as automorphism groups of planar domains. A bit more is known about which groups can be the automorphism groups of compact Riemann surfaces [HEI1].

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We begin the proof of Theorem 8.8 with the following technical lemma: Lemma 8.10 Let G be a compact subgroup of the unitary group U(n), and dim G ≥ n2 − 2n + 3. Then either G = U(n), or G = SU(n). Proof: The proof is a nontrivial exercise in basic representation theory. See [ISSK2].

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Proof of Theorem 8.8: Let p ∈ M and Ip denote the isotropy group of p in Aut(M). Since the complex dimension of M is n, the real dimension of any (local) orbit of the action of Aut(M) on M does not exceed 2n, and therefore we have dim Ip ≥ n2 − 2n + 3. Let g = gij dz i dz j be the Wu metric on M [WU]. The metric g is biholomorphically invariant and, since M is hyperbolic, defines a (positive-definite) Hermitian form on the tangent space Tp(M) to M at p [WU]. Let Up denote the group of linear transformations of Tp(M) preserving the form g(p). Consider the mapping α : Ip → Up : α(f) := df(p),

f ∈ Ip .

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The mapping α is a continuous group homomorphism and thus a Lie group homomorphism (see [WA]). By the Bochner linearization theorem (or see [EIS] and [KIE]), it is one-to-one. Further, since dim Ip ≥ n2 − 2n + 3, and Ip is compact (see [KOB1]), α(Ip) is a compact subgroup of Up of dimension at least n2 − 2n + 3. By Lemma 8.10, α(Ip ) is either Up or SUp (the latter denotes the subgroup of Up of matrices with determinant 1). The group Up acts transitively on real directions in Tp(M) and the group SUp on complex directions (see [GRK7] and [BDK1], [BDK2] for terminology). Since M is non-compact because dim [Aut(M)] > 0—see [KOB1]—the main result of [GRK7] and its generalization in [BDK1], [BDK2] applies. Thus M is biholomorphically equivalent to the ball.

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We close this subsection by noting that the domain Ω = Bn−1 × D—the product of the (n−1)-dimensional ball and the unit disc in C—has dimension n and automorphism group of dimension n2 + 2. So in this sense our theorem is sharp. And of course Ω is not biholomorphic to the ball B.

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8.3

Discussion of the Proof of Theorem 8.2 and Its Generalizations

It is convenient to prove this more general version of Theorem 8.2 (see [ISA1]): Theorem 8.11 Let M be a connected Stein manifold of complex dimension n and suppose that Aut(M) and Aut(Cn ) are isomorphic as topological groups (both groups are considered with the compact-open topology). Then M is biholomorphically equivalent to Cn . Proof: The proof will be broken up into several steps, some of them enunciated as lemmas. Let Φ : Aut(Cn ) → Aut(M) be an isomorphism. The group Aut(Cn ) contains the torus Tn , i.e., all transformations of the form

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(z1, . . . , zn ) 7→ (eiθ1 z1 , . . . , eiθn zn ),

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with θ1, . . . , θn ∈ R. Therefore Tn acts on M with the action mapping F : Tn × M → M defined as follows: F (t, p) := Φ(t)(p),

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for t ∈ Tn , p ∈ M. This action is clearly effective on M. Since F is continuous in (t, p) and holomorphic in p, it is in fact real-analytic in (t, p) [BM]. It now follows from [BBD] that there exists a holomorphic embedding α : M → Cn such that E := α(M) is a Reinhardt domain and the induced action G(t, z) = (G1 (t, z), . . . , Gn (t, z)) := α(F (t, α−1 (z))) of Tn on E has the form Gj (t, z) = ei(aj1 θ1 +···+ajn θn ) zj ,

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where z = (z1 , . . . , zn ) ∈ E and ajk are fixed integers such that det(ajk ) = ±1. Let α b denote the isomorphism between Aut(M) and Aut(E) induced by α: α b(f) := α ◦ f ◦ α−1 , f ∈ Aut(M).

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Then Ψ := α b ◦ Φ is an isomorphism between Aut(Cn ) and Aut(E). It follows from the preceding discussion that Ψ(Tn ) consists precisely of all mappings of the form (8.11.1) (z1, . . . , zn ) 7→ (eiσ1 z1 , . . . , eiσn zn ) ,

where (z1, . . . , zn ) ∈ E, σ1, . . . , σn ∈ R. Let C(Tn ) denote the centralizer of Tn in Aut(Cn ), i.e., C(Tn ) := {f ∈ Aut(Cn ) : f ◦ t = t ◦ f 27

for all t ∈ Tn }.

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Lemma 8.12 The set C(Tn ) consists of all mappings of the form (z1, . . . , zn ) 7→ (λ1 z1 , . . . , λn zn ),

(8.12.1)

where λj ∈ C \ {0}. Proof of Lemma 8.12: The proof is a simple algebraic analysis, along the lines of H. Cartan’s classic theorem about automorphisms of complete circular domains. See [KRA1, Ch. 11].

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Let C(Ψ(Tn )) denote the centralizer of Ψ(Tn ) in Aut(E). Clearly Ψ(C(Tn )) = C(Ψ(Tn )). Since Ψ(Tn ) consists precisely of all mappings (8.11.1), it follows by exactly the same argument as in Lemma 8.12 that any element of C(Ψ(Tn )) has the form (8.12.1). If we denote by H the n-dimensional complex Lie group of all mappings (8.12.1) (H is clearly isomorphic to C∗n ), then we see that Ψ induces a continuous injective homomorphism from H into itself. Since H is a Lie group, this homomorphism is in fact smooth and thus is onto, i.e., C(Ψ(Tn )) = H. In particular, E is invariant under all mappings of the form (8.12.1), and thus E is either Cn or a domain of the form

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Cn \ ∪rk=1 {zik = 0},

r ≥ 1.

(8.13)

It now remains to show that if E is of the form (8.13), then Aut(E) cannot be isomorphic to Aut(Cn ). This is a consequence of the following observation:

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Lemma 8.14 We have

(8.14.1) Aut(Cn ) is connected;

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(8.14.2) If E is a domain of the form (8.13), then Aut(E) is disconnected.

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Proof of Lemma 8.14: The proof consists of an analysis of the action of shears on Cn . Completion of the Proof of Theorem 8.11: It now follows from Lemma 8.14 that Aut(Cn ) and Aut(E) cannot be isomorphic as topological groups. Thus, M is holomorphically equivalent to Cn , and the theorem is proved.

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There is a variety of work characterizing the dimensions of automorphism groups. The paper [GIK] is particularly noted for the use of partition theory, and considerable computer power. Also the papers [ISA1], [ISA2], [ISA3] explore themes related to the ones in this section. We refer the reader to [GIK] and [ISK1] for further information about the dimensions of the automorphism groups of domains.

9

The Scaling Method

9.1

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The scaling method is a technique for, in effect, systematically distorting a domain in such a fashion as to force it to approximate a model domain. We shall motivate the scaling method by beginning with an example in one dimension.

Scaling of the Unit Disc

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We now consider scaling in perhaps its most elementary context, that of complex dimension one and the domain the unit disc. This will set the stage for our more sophisticated considerations in several complex variables which will appear later on. Let D be the open unit disc in the complex plane C. Choose a sequence bj ∈ D satisfying the conditions

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0 < bj < bj+1 < · · · < 1, ∀j = 1, 2, . . . ,

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and

lim bj = 1.

j→∞

Consider the sequence of dilations Mj (ζ) =

1 (ζ − 1) . 1 − bj

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Write λj = 1 − bj . Then ¯ < 1} Mj (D) = {ζ ∈ C | (1 + λj ζ)(1 + λj ζ) = {ζ ∈ C | 2 Re ζ < −λj |ζ|2} . 29

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We see that the sequence of sets Mj (D) converges to the left half-plane H = {ζ ∈ C | Re ζ < 0} in the sense that Mj (D) ⊂ Mj+1 (D), ∀j = 1, 2, . . . , and

∞ [

Mj (D) = H.

j=1

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[Compare the concept of convergence in the Hausdorff metric on sets—see [FED]. See also the concept of normal convergence of domains which we discuss below.] Now we combine this simple observation with the fact that there exists the sequence of maps ζ + bj ϕj (ζ) = 1 + bj ζ that are automorphisms of D satisfying ϕj (0) = bj . Consider the sequence of composite maps σj ≡ Mj ◦ ϕj : D → C. A direct computation yields that

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  1 ζ + bj −1 Mj ◦ ϕj (ζ) = 1 − bj 1 + bj ζ ζ −1 = . 1 + bj ζ

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We see that the sequence of holomorphic mappings Mj ◦ ϕj converges uniformly on compact subsets of D to the mapping σ b(ζ) =

ζ −1 . ζ +1

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This is a biholomorphic mapping from the open unit disc D onto the left half plane H. (We have in effect discovered here a means to see a version of the Cayley map by way of scaling.) We see that we have utilized the automorphisms of the disc to confirm that the disc is conformally equivalent to a certain canonical domain—namely the halfplane. This result is neither surprising nor insightful. But it is a toy version of the main result that we shall present below, and therefore should contribute to our understanding of the situation.

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9.2

A Generalization

We now expand the simple observations of the preceding subsection to yield the statement and proof of the following one-dimensional version of the Wong-Rosay Theorem [WON1] (see Theorem 3.1). Also refer back to our Theorem 3.2.

Proposition 9.1 Let Ω be a domain in the complex plane C admitting a boundary point X such that (i) there exists an open neighborhood U of X in C such that U ∩ ∂Ω is a C 1 curve,

f

and

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(ii) there exists a sequence ϕj of automorphisms of Ω and a point P ∈ Ω such that lim ϕj (P ) = X . j→∞

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Then Ω is conformally equivalent to the open unit disc. See [KRA2] for this theorem. We shall give a proof of this result, different from that in [KRA2] and in our Section 3, that illustrates the technique of scaling.

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We present the proof in several steps. Let X ∈ ∂Ω be as in the hypothesis of the proposition. Write D(X, r) = {ζ ∈ C : |ζ − X| < r}. Transforming Ω by a conformal mapping ζ 7→ eiα (ζ − X), we may without loss of generality assume the following: (a) X = 0

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(b) Ω ∩ D(X, r) = {ζ = ξ + iη : η > ψ(ξ), |ζ − X| < r} and ∂Ω ∩ D(X, r) = {ζ : η = ψ(ξ), |ζ − X| < r} for a real-valued C 1 function ψ in one real-variable satisfying ψ(0) = 0 and ψ 0(0) = 0. Step 1. The Scaling Map: Notice that the sequence ϕj (P ) now converges to 0 as j → ∞. For each j, we choose a point Qj ∈ ∂Ω that is the nearest 31

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to ϕj (P ). Since Qj also converges to 0, replacing ϕj by a subsequence if necessary, we may assume that every Qj ∈ D(X, r/4). Now, for each j, set αj (ζ) = i

|ϕj (P ) − Qj | (ζ − Qj ). ϕj (P ) − Qj

Notice that ϕj (P ) − Qj is a positive scalar multiple of the inward unit normal ϕj (P ) − Qj vector to ∂Ω at Qj . Thus converges to the inward unit normal |ϕj (P ) − Qj | vector to ∂Ω at 0. This implies that αj in fact converges to the identity map. Consequently, there exist positive constants r1 , r2 independent of j such that, for each j, there exists a C 1 function ψj (x) defined for |x| < r1 satisfying {αj (ζ)} ∩ ([−r1, r1 ] × [−r2, r2 ]) = {ξ + iη : |ξ| < r1 , |η| < r2 , η > ψj (ξ)}.

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Here we are writing ζ = ξ + iη. Furthermore, for each  > 0, there exists δ > 0 such that ψj (ξ) < |ξ| whenever |ξ| < δ for all j.

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Next let λj = |ϕj (P ) − Qj | for each j. Consider the dilation map Vj (ζ) =

ζ . λj

Then the sequence of holomorphic mappings that we want to study is

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ψj ≡ Vj ◦ αj ◦ ϕj : Ω → C .

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Before beginning the next step, we give an overview. The automorphism ϕj preserves the domain Ω but moves P to Pj = ϕj (P ) so that Pj converges to the origin. Then the affine map αj adjusts Ω so that the direction vector ϕj (P ) − Qj is transformed to an imaginary number times the normal vec|ϕj (P ) − Qj | tor. The final component Vj in the construction simply magnifies the domain αj (Ω). Step 2. Convergence of the ψj : We shall choose a subsequence from {ψj } that converges uniformly on compact subsets of Ω. Observe first that ψj (Ω) = Vj ◦ αj ◦ ϕj (Ω) = Vj ◦ αj (Ω) 32

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since ϕj (Ω) = Ω. Choosing a subsequence of the ψj , we may assume that λj < 1 for every j. Then, since Vj is a simple dilation by a positive number, and since αj (Ω) will miss a line segment E = {−iη : 0 ≤ η ≤ b} for some constant b > 0 independent of j, we see immediately that ψj (Ω) ⊂ C \ E

b Analysis of ψ(Ω): We want to establish that b ψ(Ω) = U,

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Step 3.

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for every j = 1, 2, . . . . Therefore Montel’s theorem (in particular, a family of holomorphic functions each of which omits the same two values α and β is normal) implies that every subsequence of {ψj } admits a subsequence, which we again (by an abuse of notation) denote by ψj , that converges uniformly on compact subsets of Ω. Denote by ψb the limit of this new sequence ψj .

where U ≡ {ζ ∈ C : Im ζ > 0}.

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Let  > 0 and let K be an arbitrary compact subset of Ω. We will show b that ψ(K) ⊂ C , where C ≡ {ζ ∈ C : −  < arg ζ < π + }. Since this will be true for every  > 0, that will yield the desired result. b Choose R > 0 such that ψ(K) is contained in the disc D(0, R) of radius R centered at 0. The sequence ϕj : Ω → Ω is a normal family since C \ Ω contains a line segment with positive length. Every subsequence of ϕj contains a subsequence that converges uniformly on compact subsets, since ϕj (P ) converges to X. Let g : Ω → Ω be a subsequential limit map. Then g(P ) = X. Recall that X ∈ ∂Ω. Hence the open mapping theorem yields that g(ζ) = X for every ζ ∈ Ω. Thus the sequence ϕj itself converges uniformly on compact subsets to the constant map with value X. Therefore we may choose N > 0 such that ϕj (K) is contained in a sufficiently small neighborhood of the origin for every j > N, and hence αj ◦ ϕj (K) ⊂ C for every j > N. It follows immediately that ψj (K) ⊂ C for every j > N, and consequently that b ψ(K) ⊂ C . 33

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Since K is an arbitrary compact subset of Ω and  > 0 is as small b b ) = i, since as we please, it follows that ψ(Ω) ⊂ U. We also have ψ(P ψj (P ) = Vj ◦ αj ◦ ϕj (P ) = i for every j = 1, 2, . . .. Since we have established b this result for every  > 0, we may now conclude that ψ(Ω) ⊂ U. e be an arbitrary compact subset of Step 4. Convergence of ψj−1: Let K e ⊂ C . Choose r > 0 the upper-half plane U. Then choose  > 0 so that K such that D(0, r) ∩ C ⊂ Ω ∩ D(0, r) .

Shrinking r > 0 if necessary, since αj converges to the identity map uniformly on compact subsets of C, we see that there exists N > 0 such that

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D(0, r) ∩ C ⊂ αj (Ω) ∩ D(0, r)

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for every j > N. Hence we see that ψj−1 maps K into Ω. Since Ω ⊂ C \ E, we may again choose a subsequence of ψj , which we again denote by ψj , so that ψj−1 converges to a holomorphic map, say τ : U → Ω. Since τ is holomorphic and τ (i) = P , we see that τ maps the upper-half plane U into Ω.

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Step 5. Synthesis: We are ready to complete the proof. By the Cauchy estimates, the derivatives dψj of ψj as well as the derivatives d[ψj−1] both b ) · db converge. Therefore, dψ(P τ (i) = 1. This means that ψb ◦ τ : U → U is a holomorphic mapping satisfying ψb ◦ τ (i) = i and (ψb ◦ τ )0(i) = 1. Then, by Schwarz’s lemma, one concludes that ψb ◦ τ = id, where id is the identity mapping. Likewise, the same reasoning applied to τ ◦ ψb : Ω → Ω implies that τ ◦ ψb = id. We conclude that ψb : Ω → U is a biholomorphic mapping. That completes the proof. Remark 9.2 The sequence of mappings ψj constructed above is often called a scaling sequence. It is constructed from a composition of

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(1) the automorphisms carrying one fixed interior point successively to a boundary point, (2) certain affine adjustments, and (3) the stretching dilation map. 34

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The proof given above is a good example of the scaling technique. The main thrust of the method is that the image of the limit mapping is determined solely by the affine adjustments and the dilations, while the scaling sequence converges to a conformal mapping.

10 10.1

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Remark 9.3 As observed earlier, the main result here can be proved in a much simpler way. Namely, one may conclude immediately, from the argument on the shrinking of ϕj (K) into a simply connected subset of Ω, that Ω must be simply connected. Then the conclusion follows by the Riemann Mapping Theorem. But we are trying to avoid using the Riemann mapping theorem. The goal of this argument is to provide a basis for the scaling method which can be applied to the higher dimensional case (where there is no Riemann mapping theorem). We shall see further developments in higher dimensions in subsequent sections. It may be noted that a version of the planar Riemann mapping theorem can be proved using the scaling technique. We shall not investigate that matter here.

Higher Dimensional Scaling Non-Isotropic Scaling

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We now consider complex dimension 2. The situation in dimension n is analogous but a bit more tedious with regard to notation. We first demonstrate the scaling of the complex two-dimensional ball

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B 2 = {(z1, z2) ∈ C2 | |z1 |2 + |z2|2 < 1}

at the boundary point (1, 0).

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Denote by bj a sequence of real numbers satisfying 0 < bj < bj+1 < · · · < 1 ∀j = 1, 2, . . .

and

lim bj = 1 ,

j→∞

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and let Qj = (bj , 0) for each j = 1, 2, . . . . Then consider the translation T (z1, z2) = (z1 − 1, z2). The domain T (B 2) is now defined by the inequality |ζ1 + 1|2 + |ζ2|2 < 1 or, equivalently, by Notice that the mapping

2 Re ζ1 < −|ζ1 |2 − |ζ2 |2.

ϕj (z1 , z2) =

! p 1 − |bj |2 z1 + bj , z2 1 + bj z1 1 + bj z1

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is an automorphism of B 2 satisfying ϕj (0) = (bj , 0) for every j (just calculate by hand to see that ϕj has image in the ball and is invertible). Finally consider ! z1 z2 Vj (z1, z2) = ,p λj λj

where λj = 1 − bj for each j. Imitating the one-dimensional case (see Section 9.3), we consider the scaling sequence ψj (z1, z2 ) = Vj ◦ T ◦ ϕj (z1, z2).

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Notice here that Vj is a dilation but, unlike in the one-dimensional case, it is nonisotropic in the sense that the eigenvalues are not uniformly comparable. It is a fact of life that the geometry of several complex variables, near the boundary of a domain, is nonisotropic.

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b 1, z2) ≡ lim ψj (z1, z2), and the set We now compute the limit map ψ(z j→∞

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b 2 ). A direct computation yields the following: ψ(B q     2 1 − b j 1 z1 + bj 1 −1 ,p · · z2  ψj (z1, z2) =  λj 1 + bj z1 λj 1 + bj z1 ! p 1 + bj z2 z1 − 1 = , . 1 + bj z1 1 + bj z1 36

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Taking the limit, we find immediately the limit mapping ! √ z − 1 2 z 2 1 b 1 , z2 ) = ψ(z , z1 + 1 z1 + 1

and that ψj converges to ψb uniformly on compact subsets of B 2. Observe that the map ψb : B 2 → C2 is an injective holomorphic mapping, and that its image coincides with the Siegel half space U = {(z1, z2) ∈ C2 | 2 Re z1 < −|z2|2} .

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rep

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Therefore ψb : B 2 → U is in fact a biholomorphic mapping. Refer to Figure 3 below. It should be noted that the domain U is the standard unbounded realization of the unit ball. It is a generalization of the upper half plane in one complex dimension. It is also called a Siegel domain of type two, or a Piatetski-Shapiro domain.

Figure 3: B is biholomorphic to U.

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Observe also that one can see the convergence of the sets ψj (B 2) here. A direct calculation reveals that

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ψj (B 2) = = = =

Vj Vj Vj Vj

◦ αj ◦ ϕj (B 2) ◦ αj (B 2 )
{z ∈ C2 | |z1 + 1|2 < 1 − |z2|2 }
{z ∈ C2 | 2 Re z1 < −|z1|2 − |z2|2 }

= {z ∈ C2 | 2 Re λj z1 < −λj 2 |z1|2 − λj |z2|2 } = {z ∈ C2 | 2 Re z1 < −λj |z1|2 − |z2|2 }. 37

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Since λj & 0, it follows immediately that ψj (B 2) ⊂ ψj+1 (B 2) ∀j = 1, 2, . . . and

∞ [

j=1

ψj (B 2) = U.

b 2) is in fact the limit domain In this sense, it seems sensible to say that ψ(B 2 of the sequence of domain ψj (B ). This simple example already illustrates an important aspect of the scaling technique in complex dimension two (as well as in higher complex dimensions).

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In view of the discussion of the one-dimensional scaling (see Section 9.3), the following theorem is worthy of note. Theorem 10.1 (Wong 1977, Rosay 1979) Let Ω be a bounded domain in Cn with a boundary point X ∈ ∂Ω satisfying the following:

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(i) ∂Ω is C 2 smooth and strictly pseudoconvex near X, and (ii) there exists a sequence ϕj ∈ Aut Ω and an interior point P ∈ Ω such that limj→∞ ϕj (P ) = X. Then the domain Ω is biholomorphic to the unit ball in Cn .

Normal Convergence of Sets

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10.2

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We shall present a proof of this result, which illustrates the scaling method in detail, in subsequent sections. First we shall present a detailed exposition of the background theory.

We first describe the concept of normal convergence of domains.

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Definition 10.2 Let Ωj be domains in Cn for each j = 1, 2, . . .. The seb if the following two quence Ωj is said to converge normally to a domain Ω conditions hold: (i) If a compact set K is contained in the interior (i.e., the largest open T b subset) of j>m Ωj for some positive integer m, then K ⊂ Ω. 38

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0 b (ii) If a compact subset T K lies in Ω, then there exists a constant m > 0 0 such that K ⊂ j>m Ωj .

The reason for introducing such an idea of convergence of sets is because this is what is most natural and convenient for the scaling method and normal families with source and target domains varying (compare with the idea of convergence in the Hausdorff metric—see [FED]). Proposition 10.3 If Ωj is a sequence of domains in Cn that converges norb then: mally to the domain Ω,

(1) If a sequence of holomorphic mappings ϕj : Ωj → Ω0 from Ωj to another b then its limit domain Ω0 converges uniformly on compact subsets of Ω, 0 b is a holomorphic mapping from Ω into the closure Ω of the domain Ω0 .

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(2) If a sequence of holomorphic mappings gj : Ω0 → Ωj converges uniformly on compact subsets of Ω0 , then its limit is a holomorphic mapb of Ω. b ping from the domain Ω0 into the closure Ω

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We leave the proof of this result as an exercise for the reader.

10.3

Localization

10.3.1

Local Holomorphic Peak Functions

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Definition 10.4 Let Ω be a domain in Cn . A boundary point X ∈ ∂Ω is said to admit a holomorphic peak function if there exists a continuous function h : Ω → C that satisfies the following properties: (i) h is holomorphic on Ω,

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(ii) h(X) = 1, and

(iii) |h(z)| < 1 for every z ∈ Ω \ {X}.

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Such a function h is called a holomorphic peak function for Ω at p. We call p a (holomorphic) peak point. Furthermore, we say that a boundary point X of Ω admits a local holomorphic peak function if there exists an open neighborhood U of X such that there exists a holomorphic peak function for Ω ∩ U at X (see [GAM, p. 52 ff.]). In this circumstance we call p a local (holomorphic) peak point. 39

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Proposition 10.5 Let Ω be a bounded domain in Cn with a C 2 smooth, strictly pseudoconvex boundary point X. Let B n be the unit open ball in Cn . Let η be a positive real number satisfying 0 < η < 1. Then, for every  > 0, there exists δ > 0 such that |f(z) − X| < , ∀z with |z| < η , for every holomorphic mapping f : B n → Ω with |f(0) − X| < δ. Remark 10.6 Results of this kind have appeared even as early as [GRA]. This type of localization is extremely useful.

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Proof of the Proposition: Assume to the contrary that there exist holomorphic mappings ϕj : B n → Ω satisfying the following two conditions: (a) lim ϕj (0) = X. j→∞

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(b) ∃ > 0 for which there exists a sequence zj ∈ B n such that |zj | < η and |ϕj (zj ) − X| ≥  for every j = 1, 2, . . ..

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Let U be an open neighborhood of X such that there exists a local holomorphic peak function h : Ω ∩ U → C at X. (Here we use the fact that a strictly pseudoconvex boundary point always admit a local holomorphic peak function—see [KRA1, Ch. 5].)

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Since Ω is bounded, Montel’s Theorem yields that ϕj admits a subsequence that converges uniformly on compact subsets. By an abuse of notation, we denote the subsequence by the same notation ϕj , and then the subsequential limit mapping by F : B n → Ω.

Take an open neighborhood V of 0 sufficiently small so that it satisfies the properties:

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(1) V ⊂ B n .

(2) There exists N > 0 such that ϕj (V ) ⊂ U ∩ Ω for every j > N.

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Plurisubharmonic Peak Functions

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10.3.2

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Consider the sequence of mappings h◦ϕj |V : V → D, where D is the open unit disc in C. Apply Montel’s theorem again to this sequence. Choosing a subsequence from ϕj again, we may assume that h◦ ϕj |V converges uniformly on compact subsets of V to a holomorphic map G : V → D. Since G(0) = 1 and |G(ζ)| ≤ 1 for every ζ ∈ V , the maximum principle implies that G(ζ) ≡ 1 for every ζ ∈ V . By the properties of the local holomorphic peak function h at X, this implies that F (ζ) = X for every ζ ∈ V . Since V is open, and since F is holomorphic, it follows that F (z) = X for every z ∈ B n . Since the convergence of ϕj to F is uniform on compact subsets, it is impossible to have zj with |zj | ≤ η such that ϕj (zj ) stays away from X for every j. This contradiction completes the proof.

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There is an effective method of localization in a more general setting (see [SIB]). The thrust of this method is that it avoids Montel’s theorem altogether. Thus, for instance, the hypothesis that Ω is bounded is no longer needed. Definition 10.7 Let Ω be a domain in Cn and let X be a boundary point. If there exists a continuous function h : Ω → R satisfying:

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(i) h is plurisubharmonic on Ω, and

(ii) h(X) = 0 and h(z) < 0 for every z ∈ Ω \ {0},

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then we call h a plurisubharmonic peak function at X for Ω. In such a case, X is called a plurisubharmonic peak point for Ω. Likewise, a boundary point X of the domain Ω is called a local plurisubharmonic peak-point if there exists an open neighborhood of X in Cn such that X is a plurisubharmonic peak point for Ω ∩ U. In this circumstance we call p a local plurisubharmonic peak point. We present first the following lower bound estimate for the Kobayashi metric near a local plurisubharmonic peak boundary point. See [SIB].

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Proposition 10.8 Let Ω be a bounded domain in Cn with a boundary point X ∈ ∂Ω which admits a local plurisubharmonic peak function for Ω. Then, for every open neighborhood U of X in Cn , there exists an open neighborhood V with X ∈ V ⊂ U such that the inequality kΩ (z, ξ) ≥

1 kΩ∩U (z, ξ), ∀(z, ξ) ∈ (Ω ∩ V ) × Cn , 2

where kΩ denotes the infinitesimal Kobayashi pseudo-metric of the domain Ω (see [KRA1, Ch. 11]). Proof. Denote by Dr the open disc in C of radius r centered at the origin. For the open unit disc, write D = D1 .

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By the definition of the Kobayashi metric [KRA1, Ch. 11], it suffices to prove the following statement:

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(10.8.1) It is possible to choose a neighborhood V , as specified in the proposition, so that the following holds: Given (z, ξ) ∈ (Ω ∩ V ) × Cn , every holomorphic mapping f : D → Ω from the unit disc D into Ω satisfying f(0) = z, df 0(λ) = ξ for some λ > 0 enjoys the property that f(D1/2) ⊂ U.

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Replacing U by a smaller neighborhood of X if necessary, let ψ1 : U ∩ Ω → C be a local plurisubharmonic peak function at X. Choose an open neighborhood U1 of X inside U and a constant c1 > 0 such that sup{ψ1(z) | z ∈ Ω ∩ ∂U1} = −c1 .

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Choose a neighborhood V1 of X inside U1 such that n c1 o V1 = z ∈ Ω ∩ U1 | ψ1(z) > − . 2

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Then we can extend ψ1 to a new function ψ2 : Ω → R by   if z ∈ Ω ∩ V1 ψ1 (z) ψ2 (z) = max{ψ1(z), −3c1 /2} if z ∈ Ω ∩ (U1 \ V1 )   −3c1 /2 if z ∈ Ω \ U1 . 42

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Notice that ψ2 is a global plurisubharmonic peak function for Ω at p. There is no harm in assuming (by a simple dilation) that the analytic disc f is holomorphic in a neighborhood of the closed unit disc D. Let a > 0 be such that ψ2 ◦ f(0) > −a. Consider Ea = {θ ∈ [0, 2π] | ψ2 ◦ f(eiθ ) ≥ −2a} . By the sub-mean-value inequality, we see that

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−a < ψ2 ◦ f(0) Z 1 ψ2 ◦ f(eiθ ) dθ ≤ 2π [0,2π] Z 1 ≤ (−2a) dθ 2π [0,2π]\Ea a ≤ − (2π − |Ea |), π where |S| denotes the Lebesgue measure of the set S. Hence we see that

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|Ea | > π.

Now consider a plurisubharmonic function given by υ (z) =  log kz − pk,

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where  is a certain positive constant to be chosen shortly. This is often called an anti-peak function as it satisfies υ (p) = −∞. Let inf{ψ1(z) + υ (z) : z ∈ Ω ∩ ∂V1 } = −c2, and

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sup{ψ1(z) + υ (z) : z ∈ Ω ∩ ∂U1} = −c3.

Choose  > 0 so that

−c3 < −c2 < 0.

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Extend ψ1 + υ to the plurisubharmonic function Υ : Ω → R defined by   ψ1 (z) + υ (z) if z ∈ Ω ∩ V1    c2 + c3 Υ(z) = max{ψ1(z) + υ (z), − 2 } if z ∈ Ω ∩ (U1 \ V1 )   c + c3  − 2 if z ∈ Ω \ U1 . 2 43

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Observe that Υ−1 (−∞) = {X}.

For each ζ ∈ D1/2 , apply the Poisson integral formula to obtain 1 Υ ◦ f(ζ) ≤ 10π

Z

2π 0

Υ ◦ f(eiθ ) dθ.

We now focus on the peak function ψ2 and the anti-peak function Υ. Since the sets Gk = {z ∈ Ω | ψ2(z) ≥ −1/k} for k = 1, 2, . . . form a neighborhood basis for X in Ω, we see that, for each L > 0, there exists a > 0 with a arbitrarily small such that {z ∈ Ω | ψ2(z) ≥ −2a} ⊂ {z ∈ Ω | Υ(z) < −L}.

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We have:

Claim. If a holomorphic function f : D → Ω satisfies ψ2 ◦ f(0) > −a, then Υ ◦ f(ζ) ≤ −L/10 for every ζ ∈ D1/2 .

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The proof is immediate: simply check for each ζ ∈ D1/2 that

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Z 2π 1 Υ ◦ f(eiθ ) dθ Υ ◦ f(ζ) ≤ 10π 0 Z Z 1 1 (−L) dθ + 0 dθ ≤ 10π Ea 10π [0,2π]\Ea L = − . 10

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Finally we are ready to finish the proof. Observe that the sets   k Uk = z ∈ Ω | Υ(z) < − 10

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for k = 10, 11, . . . also form a neighborhood basis for X in Ω. By the Claim above, for each k we may choose ak > 0 such that (1) Υ(z) > −k whenever ψ2 (z) > −2ak , and (2) a0 > a1 > · · · → 0. 44

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Consequently, if we choose Vk = {z ∈ Ω | ψ2 > −ak } for each k, then it follows immediately that f(0) ∈ Vk ⇒ f(D1/2 ) ⊂ Uk for every k = 10, 11, . . .. That completes the proof. Proposition 10.8 Let Ω be a bounded domain in Cn with a boundary point X ∈ ∂Ω which admits a local holomorphic peak function h for Ω. Let K be a compact subset of Ω and let P ∈ Ω. Then, for every open neighborhood U of X in Cn , there exists an open set V with X ∈ V ⊂ U such that f(K) ⊂ U whenever f : Ω → Ω is a holomorphic mapping satisfying f(P ) ∈ V .

f

Proof. Note first that a local holomorphic peak function h at X generates the local plurisubharmonic peak function log |h| at X.

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Since the Kobayashi pseudo-distance dM : M × M → R is continuous for any complex manifold M, we may select R > 0 such that the Kobayashi distance-ball BΩK (q, R) = {z ∈ Ω | dΩ (z, q) < R}

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contains K.

Then use the local holomorphic peak function h : U ∩ Ω → D at X. The distance decreasing property implies that lim

dΩ∩U (z, Qj ) ≥

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Ω∩U 3Qj →X

lim

Ω3Qj →X

dD (h(z), h(Qj )) = ∞.

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Moreover, the local holomorphic peak function and the distance-decreasing property guarantee the existence of an open set U 0 with X ∈ U 0 ⊂ U 0 ⊂ U and an open neighborhood V with X ∈ V ∈ U 0 such that dΩ∩U (z, w) > 3R

for every z ∈ V and every w ∈ ∂U 0 ∩ Ω.

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Now let ζ ∈ Ω \ U. Then, by the definition of the Kobayashi metric, there exists a piecewise smooth “almost-the-shortest” connector γ : [0, 1] → Ω with this property: We have γ(0) = z, γ(1) = ζ induced from the holomorphic chain in the definition of the Kobayashi metric such that LK Ω (γ) −

R < dΩ (z, ζ) < LK Ω (γ) . 2 45

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Here LK Ω (γ) denotes the length of γ measured by the Kobayashi metric of Ω. Since γ([0, 1]) has to cross ∂U 0 ∩ Ω, we let t ∈ (0, 1) be such that γ([0, t)) ⊂ U 0 ∩ Ω and γ(t) ∈ ∂U 0 ∩ Ω. Then it follows that 1 K 1 3 K LK Ω (γ) > LΩ (γ|[0,t]) > LΩ∩U (γ|[0,t]) > dΩ∩U (z, γ(t)) > R. 2 2 2 This therefore implies that R 3 dΩ (z, ζ) > R − = R. 2 2 In particular, BΩK (z, R) ⊂ U whenever z ∈ V .

f

Since the f given in the hypothesis is a holomorphic mapping, the distancedecreasing property of the Kobayashi distance yields that

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f(K) ⊂ f(BΩK (P, R)) ⊂ BΩK (f(P ), R) ⊂ U.

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Since f(P ) ∈ V , we see that BΩK (f(P ), R) ⊂ U. This is what we wanted to establish.

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Remark 10.9 This argument avoiding Montel’s Theorem is also useful in infinite dimensions. See [KIK4] for a treatment of normal families in the infinite-dimensional context.

Concluding Remarks

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The theory of automorphism groups of domains in Cn is a dynamic instance of group actions in complex analysis. This is an active and growing area of modern mathematics, and is a lovely meeting ground of analysis, geometry, and Lie theory. The Greene/Krantz conjecture has been a suggestive train of thought in the subject for nearly thirty years now, and will continue to be a source of ideas and techniques. We hope that this modest expository paper will inspire new workers in the field.

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Department of Mathematics Campus Box 1146 Washington University St. Louis, Missouri 63130 [email protected]

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