contentsharish/papers/auto-survey.pdf · harish seshadri, kaushal verma contents 1. introduction 1...

SOME ASPECTS OF THE AUTOMORPHISM GROUPS OF DOMAINS

HARISH SESHADRI, KAUSHAL VERMA

Contents

1. Introduction 12. Domains with noncompact automorphism groups 42.1. A theorem of Siegel 42.2. The Wong–Rosay Theorem 82.3. The theorems of Bedford–Pinchuk 92.4. The Greene–Krantz conjecture 112.5. Domains with nonsmooth boundaries 113. Domains with positive-dimensional automorphism groups 134. Miscellaneous Topics 164.1. Variation of the automorphism group with D 164.2. Determining sets for automorphisms 184.3. A characterization of Cn 194.4. Automorphisms of complements of analytic sets 19References 20Index 24

1. Introduction

For a bounded domain D ⊂ Cn, let Aut(D) denote the group of holomorphic automorphismsof D. The purpose of these notes is to provide an overview of some aspects of Aut(D) withthe intention of summarizing some of the known ways in which the intrinsic geometry of D andAut(D) influence each other. To start with, let us recall a basic theorem of H. Cartan:

Theorem 1.1. The natural action Aut(D)×D → D given by

(f, z) 7→ f(z)

is proper, real-analytic and Aut(D) is locally compact in the compact open topology. Further-more, Aut(D) has a structure of a real Lie group that is compatible with the compact opentopology and

dim Aut(D) ≤ n2 + 2n.

Recall that a continuous action of a topological group G on a locally compact Hausdorff spaceX is said to be proper if the mapping

G×X → X ×X(g, x) 7→ (gx, x)

is proper. In other words, the subset

g ∈ G : g(K1) ∩K2 6= ∅is compact in G for any pair of compact sets K1,K2 ⊂ X. Properness of a group action is adesirable property for it implies, among other things, that the quotient X/G is Hausdorff. The

1

2 HARISH SESHADRI, KAUSHAL VERMA

reason why Aut(D) acts properly on D comes from a compactness principle, due again to H.Cartan:

Cartan’s compactness principle: Every sequence φj ∈ Aut(D) contains a convergent subsequenceand the limit map φ either maps D into ∂D, i.e., φ(D) ⊂ ∂D or else φ ∈ Aut(D).

Hence the limit map is forced to be in Aut(D) if its image intersects D. Using this, it can beseen that for a given pair of compact sets K1,K2 ⊂ D, every sequence of automorphisms φj withthe property that φj(K1) ∩K2 6= ∅ contains a convergent sequence whose limit φ ∈ Aut(D). Inparticular, the isotropy subgroup Iz = φ ∈ Aut(D) : φ(z) = z is compact for every z ∈ D andas mentioned above, D/Aut(D) is Hausdorff.

A basic question regarding Aut(D) is to understand the extent to which it determines D.While it is evident that biholomorphic domains have isomorphic automorphism groups, theconverse does not hold. Indeed, the conformal type of the annulus AR = z ∈ C : r < |z| < 1is determined by r but for r 6= s, Aut(Ar) ' Aut(As) as each is generated by rotations z 7→ eiθz

(' S1) and the involution z 7→ r/z. Thus, it is natural to ask for additional hypotheses onAut(D) that can potentially determine D up to biholomorphism.

Broadly speaking, this article is organized as follows. In Sections 2 and 3, we survey resultsabout domains with ‘large’ automorphism groups. There are two notions of ‘largeness’ whichhave been extensively studied: we can require Aut(D) to be (i) non-compact or (ii) a positive-dimensional Lie group. In Section 4 we briefly survey general results about automorphisms ofdomains. In more detail we proceed as follows:

We begin Section 2 by giving a detailed proof of a classical result of C. L. Siegel. To motivatethis, we note that a paradigm for studying Aut(D) is provided by the symbiotic relationshipbetween D and D/Aut(D), which manifests in a variety of ways. A basic problem is to under-stand the influence of D/Aut(D) on D. Suppose that D/Aut(D) is compact. This condition issatisfied in at least two important cases, namely, when D is homogeneous and when there is adiscrete subgroup Γ ⊂ Aut(D) with compact fundamental region. It turns out that compactnessof D/Γ implies that D must be a domain of holomorphy. In particular, bounded domains thatcover compact complex manifolds must be domains of holomorphy. Of the several ways of seeingthis, we discuss C. L. Siegel’s proof.

The case when Aut(D) is noncompact naturally brings the boundary ∂D into considerationfor then there is an orbit in D that accumulates at a boundary point by Cartan’s compactnessprinciple. A prototype theorem here is the B. Wong–J.-P. Rosay theorem ([67], [53]), accordingto which a C2-smooth strongly pseudoconvex domain D for which Aut(D) is non-compact mustbe biholomorphic to the unit ball Bn. In fact, the hypotheses can be weakened considerably – Dis biholomorphic to Bn even if a single orbit accumulates at a strongly pseudoconvex boundarypoint. In particular, the boundary ∂D need not even be globally smooth and in this case,local boundary information at an orbit accumulation point propagates to determine the domaincompletely. This phenomenon has been systematically studied in several cases by E. Bedford–S.Pinchuk ([3], [4], [5], [6]), F. Berteloot ([8], [9]), H. Gaussier ([23]) and K.-T. Kim ([42], [43])among others and the first reduction in this problem is afforded by utilizing Pinchuk’s rescalingmethod. This technique requires an a priori knowledge of what the boundary is like near anorbit accumulation point and it essentially consists of blowing up an infinitesimal neighbourhoodof a boundary point, by an ever increasing sequence of dilations, to reveal its intrinsic geometricstructure. On the other hand, S. Frankel [17] introduced another intrinsic way to rescale a convexdomain (in the usual geometric sense) without any smoothness assumptions on the boundary.We discuss both methods with an emphasis on the central concepts and their wide applicabilityin Sections 2.3, 2.4 and 2.5. A relevant question here is

SOME ASPECTS OF THE AUTOMORPHISM GROUPS OF DOMAINS 3

The Greene–Krantz conjecture: For a smoothly bounded domain D, every orbit that accumulatesat ∂D does so only at points of finite type.

Though this is open in full generality, it has been verified in several cases by K.-T. Kim andothers. A discussion of what’s known about this question forms the content of Section 2.4.

It is relevant and important to mention early work of J. Vey [61], [62] that deals with compactquotients of Siegel domains and its analogue for convex sets in Rn with affine transformationsplaying the role of automorphisms. Further recent work along these lines deals with domainsin projective spaces and the problem of classifying them under suitable hypotheses on theirautomorphism groups.

In Section 3, we study domains with positive-dimensional automorphism groups. A funda-mental question one can ask is which real Lie groups arise as automorphism groups of boundeddomains in Cn. E. Bedford - J. Dadok [2] and R. Saerens–W. Zame [55] showed that it is possibleto realise every connected, compact real Lie group as the full automorphism group of a smoothreal-analytic strongly pseudoconvex domain in CN for some large N . Recent work of J. Winkel-mann [64] shows that any connected real Lie group can be realised as the full automorphismgroup of a Kobayashi complete hyperbolic Stein manifold such that every orbit is totally real.Using the canonical Grauert tube in the tangent bundle of the given real Lie group, S. J. Kan[40] showed that the Stein manifold, with all of the above properties, can be chosen so that itscomplex dimension equals the real dimension of the given Lie group.

The second question that can be asked is this: for a given integer i, 0 ≤ i ≤ n2+2n, classify allD such that dim Aut(D) = i. That the top dimension is achieved only by domains biholomorphicto the ball was shown by W. Kaup [41] and much recent work along this line is due to A. V.Isaev [32], [33] and A. V. Isaev–S. G. Krantz [36]. The main result in [36] shows that there areno Kobayashi hyperbolic domains D ⊂ Cn (n ≥ 2) for which

n2 + 3 ≤ dim Aut(D) ≤ n2 + 2n.

Further work of Isaev gives a complete classification of homogeneous domains D ⊂ Cn (n ≥ 2)for which

n2 − 1 ≤ dim Aut(D) ≤ n2 + 2

and those D (without any homogeneity assumptions) for which dim Aut(D) = n2 − 1. Sincethese have been described in the monograph [34] by Isaev, we refer the reader to it for details.For lower dimensions, the problem remains understandably untractable for this amounts toidentifying D with fewer symmetries. Along similar lines, it is reasonable to believe that thetopological complexity of a domain has an adverse effect on the dimension of its automorphismgroup. This was made precise by Bedford [1] who considered pseudoconvex domains with nontrivial topology. He showed that if D is a bounded pseudoconvex domain with Hk(D,Z) 6= 0,then

dim Aut(D) ≤ 2n+ k(k + 1)/2 + (n− k)2

and that if Hn(D,R) ≥ 2, then

dim Aut(D) ≤ n(n+ 3)/2.

Given the ubiquity of a topic such as this one, the existence of several interesting themesthat do not perhaps fit in the framework described above is inevitable. For example, thereare questions related to the upper semicontinuity of the automorphism group as a function ofthe domain. Work of R. Greene–S. Krantz–K.-T. Kim–A. Seo ([26], [27]) B. L. Fridman–E. A.Poletsky [22] among others is relevant here. Then there are questions about determining sets forautomorphisms studied by Fridman–Kim–Krantz–Ma [18] and a characterization of Cn in termsof its large automorphism group due to G. Buzzard–S. Merenkov [11] and Isaev [35]. We also


discuss some work of Winkelmann [65] on the automorphisms of the complements of analyticsets–this is relevant to Bedford’s theorem mentioned above in the sense that it illustrates theinfluence of topological complexity on the automorphism group. We conclude by reporting onsome recent work of A. Zimmer [68], [69].

The authors would like to thank the referee for very carefully reading an earlier draft of thisarticle. The suggested changes and comments have all helped in improving it considerably.

2. Domains with noncompact automorphism groups

The main assumption in this section is the noncompactness of Aut(D). On the other hand,we do not require that Aut(D) has positive dimension, i.e., Aut(D) is allowed to be discrete.

2.1. A theorem of Siegel. Let Γ ⊂ Aut(D) be a discrete subgroup. Cartan’s compactnessprinciple implies that Γ acts properly discontinuously on D, i.e., every z ∈ D has a neighbour-hood U so that γ ∈ Γ : γ(U)∩U 6= ∅ is finite. The purpose of this section is to give a completeproof of the following theorem of C. L. Siegel ([58]):

Theorem 2.1. If D/Γ is compact then D is a domain of holomorphy.

The proof we present is due to Siegel. While different proofs of this fact have appeared inlater expositions, we feel that the original proof is interesting and deserves to be presented indetail.

Each γ ∈ Γ is a self-map of D and therefore its complex Jacobian determinant jγ(z) is welldefined. We then have the following:

Lemma 2.2.

(2.1)∑γ∈Γ

|jγ(z)|2

converges uniformly on compact subsets of D.

To see this, fix a compact C ⊂ D and let 2r be the euclidean distance between C and ∂D.Let Cr ⊂ D be an r-thickening of C, i.e., Cr consists of those points in D that are at a distanceat most r from C. Note that Cr ⊂ D is compact and that for each a ∈ C, the ball B(a, r) ⊂ Cr.For brevity, write ω = Vol(B(a, r)) and let m be the cardinality of γ ∈ Γ : γ(Cr) ∩ Cr 6= ∅.For a ∈ C,

|jγ(a)|2 ≤ ω−1

∫B(a,r)

|jγ(z)|2 dVz = ω−1Vol(γ(B(a, r))

where the first inequality follows by the mean value property and the second is a consequenceof the fact that the real Jacobian of γ is |jγ(z)|2. Therefore∑

γ∈Γ

|jγ(a)|2 ≤ ω−1∑γ∈Γ

Vol(γ(B(a, r)).

For a given γ ∈ Γ, at most m sets of the form γ′(B(a, r)), γ′ ∈ Γ overlap with γ(B(a, r)) andhence the collection of sets γ(B(a, r)) : γ ∈ Γ cover any point of D at most m times. Hence∑

γ∈Γ

|jγ(a)|2 ≤ mω−1Vol(D),

which shows that (2.1) converges pointwise on C. To show that the convergence is uniform onC, let K be a larger compact set containing C such that Vol(K \ C) ≤ ε. Again, only finitelymany of the sets γ(C) : γ ∈ Γ intersect K. By ignoring such terms and following the samesteps as above, we get ∑′

|jγ(a)|2 ≤ mVol(K \ C) ≤ mε


where the prime on the sum emphasizes that only those γ ∈ Γ are being considered for whichγ(C) ∩K = ∅. This holds for every a ∈ C and shows that the convergence in (2.1) is uniformon C.

We can now prove Theorem 2.1. Suppose that Γ has a compact fundamental region, sayF ⊂ D. To show that D is a domain of holomorphy it suffices to show that D can be exhaustedby an increasing sequence of analytic polyhedra (see ,or example, Theorem 2.1 of [13]). To thisend, let C ⊂ D be compact and assume without loss of generality that F ⊂ C. Since (2.1)converges uniformly on C, there exists M > 0 be such that |jγ(z)| < M for all γ ∈ Γ and z ∈ C.In fact, by taking γ = e, the identity in Γ, we see that M > 1. Also, |jγ(z)| < 1/M for all z ∈ Cand for all but finitely many γ. Said differently,

|jγ−1(z)| < 1/M

for all z ∈ C except for finitely many γ, say γ1, γ2, . . . γr. Let

C1 = γ−11 (C) ∪ γ−1

2 (C) ∪ · · · ∪ γ−1r (C).

Note that e is certainly amongst γ1, γ2, . . . , γr and hence C ⊂ C1. Let C2 ⊂ D be a relativelycompact domain such that C1 ⊂ C2. Since F ⊂ D is the fundamental domain of Γ, it followsthat

C2 ⊂⋃γ∈Γ

γ(F )

but Cartan’s compactness principle implies that only finitely many of γ(F ) actually intersect C2.Hence C2 is covered by the union of only finitely many sets of the form γ(F ) and therefore byfinitely many of the form γ(C) since F ⊂ C. Of these, γ−1

1 (C), γ−12 (C), . . . , γ−1

r (C) are certainlythere since their union is C1, which is a subset of C2. Let

C2 \ C1 ⊂ γ−1r+1(C) ∪ γ−1

r+2(C) ∪ · · · ∪ γ−1r+s(C).

Thus a given z ∈ C2 \C1 can be expressed as z = γ−1z0 for some γ ∈ γr+1, γr+2, . . . , γr+s andz0 ∈ C. For such a γ, the chain rule shows that

jγ(z)jγ−1(z0) = 1

and since |jγ−1(z0)| < 1/M , it follows that |jγ(z)| > M . Putting all this together we get thatfor z ∈ C2 \ C1

|jγr+k(z)/M | > 1

for some k, 1 ≤ k ≤ s, while for z ∈ C

|jγr+k(z)/M | < 1

for all k, 1 ≤ k ≤ s. It follows that the analytic polyhedron defined as

P =z ∈ C2 : |jγr+1(z)/M | < 1, |jγr+2(z)/M | < 1, . . . , |jγr+s(z)/M | < 1

is such that C ⊂ P ⊂ C1. This shows that D can be exhausted by an increasing sequence ofsuch analytic polyhedra. Hence D must be a domain of holomorphy.

In fact, Siegel proves more:

Theorem 2.3. D admits n independent automorphic functions relative to Γ and there exists aholomorphic function on D that blows up at each point of ∂D.

The second assertion of this theorem gives another proof that D is a domain of holomorphywhile the first assertion implies the following

Corollary 2.4. If D covers a compact complex manifold M , then M must be projective algebraic.


The proof of Theorem 2.3 proceeds as follows. Let φ be a bounded holomorphic function onD and m > 1 an integer. The Poincare series associated to φ is

(2.2) h(z) =∑γ∈Γ

φ(γz)jmγ (z),

and this is well-defined by arguments used earlier. In fact, it can be checked that

h(γz) = j−mγ (z)h(z)

for each γ ∈ Γ and thus h(z) is an automorphic form of weight m relative to Γ. Let M⊂ Γ bethe subgroup consisting of measure preserving transformations, i.e.,

M = γ ∈ Γ : |jγ(z)| = 1 on D.

Since (2.1) converges uniformly on compact subsets of D,M is a finite group of order, say r ≥ 1.Then every γ ∈M satisfies γr = e and hence jrγ(z) = 1 on D. Let M = γ1 = e, γ2, . . . , γr.

Step 1: There exists a point z0 ∈ D such that |jγ(z0)| < 1 for every γ ∈ Γ \M. In fact, the setof such points is open in D.

To see this, pick an arbitrary p ∈ D and consider its orbit Γp. Since (2.1) converges, thereare only finitely many γ ∈ Γ for which |jγ(p)| ≥ 1. Let γ be such that

|jγ(p)| = max|jγ(p)| : γ ∈ Γ

and let q = γ(p). Now for any γ ∈ Γ,

|jγ(q)| = |jγ(γ(p))| = |jγγ(p)/jγ(p)| ≤ 1

where the last inequality follows by the definition of γ and hence q has the property that|jγ(q)| ≤ 1 for all γ ∈ Γ. Again, the convergence of (2.1) implies that the number of thoseγ ∈ Γ for which |jγ(q)| = 1 is finite. Evidently, every γ ∈ M has this property. But there maybe others too and to take them into account, suppose that γr+1, γr+2, . . . , γs are the additionalones, i.e.,

|jγi(q)| = 1

for 1 ≤ i ≤ s and that |jγ(q)| < 1 for every other γ ∈ Γ. Fix a neighbourhood U of q in D. Byenumerating the elements of Γ as γi, it follows that there exists a t > s such that

(2.3)∑k>t

|jγk(z)|2 < 1/2

for every z ∈ U . By shrinking U if needed, we can assume that

(2.4) jγs+1(z), jγs+2(z), . . . , jγt(z)

are all less than 1 in modulus in U . If |jγs(z)| were to have a minimum value of 1 in U , themaximum principle applied to 1/jγs(z) would imply that it is a constant, in fact a unimodularconstant since |jγs(q)| = 1 and hence γs ∈ M, which is false by assumption. Hence there existsa point q near q such that |jγs(q)| < 1.

It follows that the orbit Γq then has at most s − 1 points, say qi = γi(q), 1 ≤ i ≤ s − 1 forwhich

|jγi(q)| ≥ 1.

Indeed, none of γk : k > t qualify due to (2.3) and (2.4) disqualifies the finite list γs+1, . . . , γtas well and the remaining candidate γs is such that |jγs(q)| < 1 by construction. The conclusionof all this is that if there are s elements in Γ whose Jacobian at q is 1 in modulus, there areplenty of nearby points with at most s − 1 elements in Γ with this property. By continuing inthis way, we see that there exists a point z0 in U such that |jγ(z0)| < 1 for every γ ∈ Γ \M.


To see that the set of such points as z0 above is open, assume that there are points zk → z0

and γk ∈ Γ such that |jγk(zk)| ≥ 1 and γk ∈ Γ \M. By Cartan’s compactness principle, either

a subsequence of γk converges to, say γ ∈ Γ \ Γ ⊂ Aut(D) or the γk’s diverge to ∂D. In thelatter event, |jγk(z)| → 0 uniformly on compact subsets of D. But this cannot happen due tothe assumed lower bound on the Jacobians of the γk’s. Hence the former possibility holds andin particular, |jγ(z0)| ≥ 1.This implies that |jγ(z)| & 1/2 near z0 and therefore |jγk(z)| & 1/4for z near z0 and for all large k. This contradicts the fact that (2.1) converges uniformly oncompact subsets of D. The only redeeming possibility is that the γk’s are eventually the same,i.e., γ = γk for all large k and this means that γ ∈ Γ. Since |jγ(z0)| ≥ 1, it follows that γ ∈ Mand hence γk ∈M for large k. This is a contradiction.

Step 2: There exist n analytically independent automorphic functions on D relative to Γ.

Take a z0 ∈ D as in Step 1. Since the fixed point set of any γ ∈ Γ other than e has dimensionat most n − 1, we may move z0 a little bit to ensure that it is not fixed by any element in M.Without loss of generality, let such a z0 be the origin; thus none of γ2, . . . , γr fix the origin and

|jγk(0)| ≤ C < 1, k ≥ r + 1

in some fixed neighbourhood of the origin. With this observation, it follows that

h(z,m) =∑γ

φ(γz)jmrγ (z),

which is a well-defined holomorphic function on D by (2.2), satisfies

limm→∞

h(z,m) = φ(γ1z) + φ(γ2z) + . . .+ φ(γrz)

uniformly near the origin. Note that this holds for all bounded functions φ on D.

In particular, by taking φ(z) = 1/r, and writing

h0(z,m) =1

r

∑γ

jmrγ (z)

we havelimm→∞

h0(z,m) = 1

uniformly near the origin. Let Φ(z) be a polynomial such that

Φ(0) = 1, Φ(γ2(0)) = . . . = Φ(γr(0)) = 0

and define Φk(z) = zkΦ2(z) for 1 ≤ k ≤ n. If

hk(z,m) =∑γ

φk(γz)jmrγ (z), 1 ≤ k ≤ n

then

fk(z,m) =hk(z,m)

h0(k,m)approaches

ψk(z) = φk(γ1z) + φk(γ2z) + . . .+ φk(γrz)

as m→∞, the convergence being uniform near the origin. Hence the Jacobian of the map

(2.5) z 7→ (f1(z,m), f2(z,m), . . . , fn(z,m))

at the origin converges to that of the map z 7→ (ψ1(z), ψ2(z), . . . , ψn(z)). The definition of theψk’s shows that

∂ψi∂zj

(0) = δij

and hence the Jacobian of the map (2.5) at the origin does not vanish for all large m.


Step 3: Let D/Γ be compact. Then there exists a holomorphic function on D that blows upnear every point p ∈ ∂D. In fact,

h0(z) = h0(z,m) =1

r

∑γ

jmrγ (z)

has this property provided m is large enough.

To see this, let pk → p ∈ ∂D. Let pk = γk(qk) for some qk ∈ C, a compact set in D andγk ∈ Γ – note that the enumeration γk is completely independent of what was done in theearlier step. Let qk → q ∈ C. By the Cauchy estimates,

|γ(z)− γ(a)| < ε

for all γ ∈ Γ and a, z that are near q provided |a − z| is small enough. Choose a such thath0(a) 6= 0. For large k, we have

|γk(a)− γk(qk)| < ε,

which when combined with |pk − p| = |γk(qk) − p|, shows that γk(a) → p. As a result, a givenelement of Γ occurs at most finitely many times in the sequence γk. By the convergence of(2.1), it follows that jγk(a)→ 0 and hence

h0(γk(a)) = j−mrγk(a) h0(a)

blows up as k →∞.

The case when D \ Γ is quasiprojective is due to N. Mok–B.Wong [51].

2.2. The Wong–Rosay Theorem. This is a theorem that characterizes Bn as the only smoothlybounded strongly pseudoconvex domain (up to biholomorphism) in Cn that has a noncompactautomorphism group.

Let D ⊂ Cn be a bounded domain. Suppose that an orbit of Aut(D) accumulates at a C2-smooth strongly pseudoconvex boundary point. Then D is biholomorphic to Bn.

B. Wong’s original result assumed that the entire boundary of D is strongly pseudoconvexand the proof relied on analysing a biholomorphic invariant associated to a given domain, theinvariant being the ratio of the Caratheodory and the Eisenman–Kobayashi volume forms. Putdifferently, for a bounded domain D ⊂ Cn and p ∈ D, it is given by

%(p) = sup|j(g f)(0)|,

the supremum being taken over the Jacobians of all possible compositions gf , wheref : Bn → Dwith f(0) = p and g : D → Bn with g(p) = 0 are holomorphic maps. Then % is a biholomorphicinvariant and hence it is constant on the orbits of Aut(D). Further, the Schwarz lemma showsthat % ≤ 1 always. Wong’s observation [67] was that if % = 1 at some point in D, thenD ' Bn. J.-P. Rosay [53] realized that no hypotheses are required on ∂D away from a C2-smooth strongly pseudoconconvex boundary accumulation point by localizing this invariant nearstrongly pseudoconvex points.

Given this characterization of the ball, it is therefore interesting to ask for a classificationof bounded domains that admit a noncompact automorphism group. The next natural step isto focus on (weakly) pseudoconvex domains with this property. Note that the orbits cannotconverge to strongly pseudoconvex points, else the domain would be equivalent to the ball. Inthis case, % < 1 on all the orbits that accumulate near pseudoconvex boundary points. Inparticular, % cannot detect the domain up to biholomorphism for any other value than 1. Othermethods are therefore needed to address this problem and this is where Pinchuk’s rescaling


technique [52] is successful. In its simplest form, consider the planar domain U whose boundaryis locally smooth near the origin and given by

r(z, z) = 2iy +O(|z|2) = (z − z) +O(|z|2).

Let pj = −iδj be a sequence of points on the inward real normal to ∂U converging to the origin.Noe that δj = dist(0, pj) = |pj |. Consider the sequence of dilations

Tj(z) = z/δj .

Then T (pj) = −i for all j ≥ 1. Also, if Uj = Tj(U), then

rj = r T−1j = δj(z − z) + δ2

jO(|z|2)

is its defining function. This degenerates as j →∞ and to avoid this, note that

rj = (1/δj)rj = (z − z) + δjO(|z|2)

is another defining function. It follows that the remainder term vanishes in the limit uniformlyon compact sets of the plane and hence

rj → r∞ = z − z = 2iy,

which defines a half plane. Thus the dilations Tj enlarge infinitesimally small neighbourhoodsof the origin to yield a limiting domain whose form is dictated by the lower order terms in r.For smooth domains in Cn, linear isotropic dilations will again give a half space in the limit as acalculation shows. To obtain limits that reflect the geometry of the boundary, S. Pinchuk usednon-isotropic dilations. For simplicty, let us consider a strongly pseudoconvex domain D in C2

with a normalised defining function as

r(z, z) = 2<z2 + |z1|2 + higher order terms.

For points of the form pj = (0,−δj) converging to the origin, the dilations

Tj(z1, z2) =(z1/√δj , z2/δj

)map pj to (0,−1) for all j and the defining functions for the domains Dj = Tj(D) converge to

(z1, z2) : 2<z2 + |z1|2 < 0in the limit. This is exactly the unbounded realization of the ball and this reflects the Levigeometry of a strongly pseudoconvex point. For any other sequence pj that converges to theorigin, the first step is to rotate coordinates so that pj lies on the inner normal and then applythe dilations. For a domain D that has an orbit φj(p) clustering at a strongly pseudoconvexpoint, say the origin, the scaling is done with respect to the sequence φj(p). The maps φj arethemselves dilated and this scaled family is shown to be normal. The limit map provides abiholomorphism between D and the ball. This proves the Wong–Rosay theorem.

It must be noted that the Wong–Rosay theorem continues to hold even when D is unbounded– see [14].

2.3. The theorems of Bedford–Pinchuk. Bedford–Pinchuk [3] considered the problem ofclassifying smoothly bounded finite type pseudoconvex domains D ⊂ C2 with noncompact au-tomorphism group. An initial scaling shows that

D ' Ω = (z1, z2) : 2<z2 + P (z1, z1) < 0where P is a real polynomial whose degree is at most the type of ∂D at the orbit accumulationpoint. Let f : D → Ω be the biholomorphism. Note that Ω admits the one parameter group ofautomorphisms Lt(z1, z2) = (z1 +z2 + it), t ∈ R. Pulling this back to D gives ht = f−1 Lt f ∈Aut(D). Then ht is shown to be parabolic in the sense that there is a q ∈ ∂D such that

limt→±∞

ht(z) = q


for all z ∈ D. Furthermore, each ht extends smoothly to D and hence

H(z) =dhtdt

(z)

∣∣∣∣t=0

is a holomorphic tangent vector field for ∂D. The existence of such a vector field on ∂Dimposes conditions on the defining function and in particular on P . It can be shown thatP = |z1|2m + other terms and a final scaling of D along the parabolic orbit (that converges to qin forward and backward time) shows that D is biholomorphic to the ellipsoid

Em = (z1, z2) ∈ C2 : |z1|2m + |z2|2 < 1,which is the bounded realisation of

(z1, z2) ∈ C2 : 2<z2 + |z1|2m < 0.The local version of this problem, in which D could possibly be unbounded and not globallysmooth but only so near the orbit accumulation point, was considered by F. Berteloot [8], [9].In this case, D is biholomorphic to its homogeneous model at the orbit accumulation point, i.e.,to

(z1, z2) ∈ C2 : 2<z2 + P (z1, z1) < 0where P is a homogeneous subharmonic polynomial of degree 2m without harmonic terms.Bedford–Pinchuk [5] also studied the convex finite type case where D was shown to be biholo-morphic to a model domain defined by a weighted homogeneous polynomial. The analogouslocal case is due to Gaussier [23].

Finally, Bedford–Pinchuk [6] were able to remove the pseudoconvexity hypothesis completelyfor domains in C2. Their classification theorem states that a bounded domain D with smoothreal-analytic boundary and noncompact automorphism group is biholomorphic to Em as above.Here, by a smooth real-analytic domain we mean a domain D whose boundary ∂D admits adefining function ρ which is real analytic in some neighbourhood of ∂D and satisfies dρ 6= 0on ∂D. Thus, in the Bedford–Pinchuk result [6], pseudoconvexity of D is not an assumptionbut a conclusion. A local version of this can be found in [59], the hypothesis there being theexistence of an orbit that accumulates at a boundary point near which the boundary is smoothreal-analytic and of finite type. Apart from this and the boundedness of D, no other assumptionsare made. It turns out that there are several possibilities for the model domains in this situation.The basic outline of the proof is to obtain model domains for each possible dimension of Aut(D)– recall that Aut(D) is at most 8-dimensional since D ⊂ C2. Using the scaling method, itis first shown that dim Aut(D) ≥ 1 and further work is required to show that Aut(D) is atleast 2-dimensional. Note that if dim Aut(D) ≥ 5, then Kaup’s result [41] shows that D ishomogeneous. In this situation, the boundary ∂D must have strongly pseudoconvex points nearthe given orbit accumulation point and by appealing to the Wong–Rosay theorem, it followsthat D must be biholomorphic to the ball. Thus, the bulk of the proof lies in obtaining modeldomains when dim Aut(D) = 2, 3 or 4. In the two-dimensional case, it is possible to identifythe normal forms of the automorphisms, which then leads to a description of the polynomialsdefining the model domains. There are in fact three model domains when dim Aut(D) = 2.When dim Aut(D) = 3, 4, Isaev’s classification in [32], [33] is checked case-by-case to identify theone that admits a bounded realisation as a domain with a smooth real-analytic finite type orbitaccumulation point. There is a unique model domain in each of the cases dim Aut(D) = 3, 4.

Apart from the smoothly bounded domains, there are nonsmooth domains as well that havenoncompact automorphism groups. For example, a polydisc in Cn has this property. One maytherefore try and classify analytic polyhedra that have this property. The non-smooth boundarypoints that arise in this case are still amenable to the scaling method as shown by K.-T. Kimamong others ([43], [48]) and two illustrative theorems in this line of thought are as follows. One,


a bounded convex domain in Cn+1 with piecewise smooth Levi flat boundary and noncompactautomorphism group is biholomorphic to the product of the disc D ⊂ C and a convex domainin Cn. In particular, this characterizes the bidisc in C2. Second, by dropping the convexityhypothesis, it is possible to show that such an analytic polyhedron in C2 must be covered bythe bidisc.

Other versions of this problem have also been considered. These include:

• the classification of domains that sit in complex manifolds, almost complex manifoldsand more generally, a Hilbert space and have a noncompact automorphism group ([60],[24], [12], [46]),

• the classification of domains in Cn with a noncompact isometry group of the Kobayashimetric - this is motivated by the question of whether every Kobayashi isometry of asmoothly bounded domain is necessarily holomorphic or conjugate holomorphic ([56],[57], [45], [25]),

• the classification of domains that have finitely smooth boundaries and a noncompactautomorphism group ([37], [38]).

2.4. The Greene–Krantz conjecture. One of the central themes in the above discussions wasto classify domains with an additional hypothesis on the boundary near the orbit accumulationpoint. For smoothly bounded domains, the extant theorems show that finite type suffices. It wasconjectured by Greene–Krantz that this must always hold. More precisely, if D is a smoothlybounded domain in Cn such that Aut(D) is noncompact, and p ∈ ∂D is an orbit accumulation,then p must be a point of finite type (in the sense of Kohn/D’Angelo/Catlin).

To start with, Kim’s theorem [43] on characterizing analytic polyhedra provides evidence forthis conjecture. Indeed, if an orbit accumulates at a smooth Levi flat point, then D must globallybe a product domain, i.e., it does not have smooth boundary. Further work in this direction isdue to Kim–Krantz [47] where they show that in C2, boundary accumulation points cannot beof (exponential) infinite type. Again, scaling methods play a crucial role here.

It is evident that further progress on this question will come from a better understanding ofpoints of infinite type.

2.5. Domains with nonsmooth boundaries. Domains with nonsmooth boundaries occurnaturally in various contexts. In particular, if D is the universal cover of a compact complexmanifold or is homogenous with respect to Aut(D), then ∂D cannot be everywhere C2-smoothunless D is biholomorphic to Bn. This follows from the Wong-Rosay theorem. Indeed, if ∂Dis C2-smooth then there exists q ∈ ∂D such that ∂D is strongly convex in a neighbourhood ofq. On the other hand, if p ∈ D then the closure (in Cn) of the orbit Aut(D)p will contain ∂D.Hence we can find a sequence γj ∈ Aut(D) such that γj(p) → q and the Wong-Rosay theoremapplies.

Examples of domains covering compact complex manifolds are bounded symmetric domainsand universal covers of Kodaira fibrations. That the latter class consists of domains was shownby P. Griffiths [31] using the Bers Uniformization Theorem [7].

Unfortunately, there are very few tools to understand the boundary behaviour of automor-phisms of domains that are not sufficiently smooth. In this section, we discuss two resultsconcerning such domains. For both of these the assumption of convexity is crucial. We beginwith a theorem of S. Frankel [17]:

Let D ⊂ Cn be a convex bounded domain covering a compact complex manifold. Then D is abounded symmetric domain


Frankel’s proof involves two main steps. The first step involves a rescaling argument to showthat Aut(D) is not discrete. The second uses Lie theory and differential geometry to completethe proof. The conclusions of this part are now known to hold in more general settings [16] andD being a domain plays only a minor role.

In light of the paucity of tools in the nonsmooth case, as mentioned above, it is worthwhileto examine the rescaling argument in detail. Throughout this section we will assume that Dis a bounded convex domain. Let K ⊂ D be compact, pj ∈ K and γj ∈ Aut(D) such thatxj = γj(pj)→ q ∈ ∂D. Define affine transformations Aj by

Aj(z) = (dγj(pj))−1(z − xj).

Frankel then shows that:

(1) The embeddings wj = Aj γj converge uniformly on compact sets to an embeddingw : D → Cn. Moreover the convex domains wj(D) converge in the Hausdorff sense to theconvex domain w(D).

(2) Suppose, in addition, that xj satisfies the following conditions: there is an affine complexline L ⊂ Cn such that q ∈ L ∩ ∂D and

(i) D = L ∩D has a unique supporting line l ⊂ L at q.(ii) xj ∈ L and the xj lie in a real affine line through q and perpendicular to l.Then the rescale blow-up w(D) contains a real affine line.

Now assume that a discrete subgroup Γ ⊂ Aut(D) acts freely and cocompactly on D. Theconvexity of D implies that almost every point of ∂D has a unique supporting hyperplane. Letq ∈ ∂D be one such point. One can then check, using the cocompactness of the action of Γ,that we can find a compact set K ⊂ D, pj ∈ K and γj ∈ Γ such that xj = γj(pj) satisfies (2)above. As the limit domain w(D) is invariant under an action of R by translations, it followsthat dim Aut(D) > 0.

The second result is due to K.-T. Kim [44]:

The Bers embedding of Teichmuller space Tg of Riemann surfaces of genus g ≥ 2 is notconvex.

The proof is based on the fact that the automorphism group of Tg is discrete. On the otherhand, using a theorem of C. McMullen [49] on the density of maximal cusps in the Bers boundary,one can show that every boundary point is an orbit accumulation point. In particular, pointswith sphere contact from inside, which exist by the convexity of the domain, can be realizedas orbit accumulation points. It turns out that the Pinchuk rescaling technique, discussedearlier, can be reformulated for sequences approaching such boundary points. Applying such arescaling one gets a limit domain that is biholomorphic to Tg and has a 1-parameter subgroupof automorphisms as in Frankel’s approach. This contradiction completes the proof.

We now explain the Kim–Krantz–Pinchuk rescaling technique in some detail: Let D be abounded convex domain and q ∈ ∂D. Let pj ∈ D be a sequence converging to q. For each jlet qj ∈ ∂D be a point with

‖pj − qj‖ = infx∈∂D

‖pj − x‖.

We can find a unitary transformation Tj : Cn → Cn such that the affine map ψj : Cn → Cndefined by ψj(z) = Tj(z − qj) satisfies

ψj(D) ⊂ (z1, . . . , zn) : Re(zn) > 0.


Denote by V jn−1 the orthogonal complement in Cn of the line joining the origin and ψj(pj) and

let Djn−1 be the projected slice

Djn−1 = z ∈ V j

n−1 : z + ψj(pj) ∈ ψj(D).

Note that Djn−1 is a domain in V j

n−1 containing the origin. Let xjn−1 be a point in ∂Djn−1 closest

to the origin. Let V jn−2 denote the orthogonal complement in V j

n−1 of the line spanned by xjn−1and let

Djn−2 = Dj

n−1 ∩ Vjn−2.

We continue this process as long as it is possible to do so.

By this process we obtain mutually orthogonal vectors xj1, . . . , xjn−1. Let xjn = ψj(pj). Then

the vectors

ejl =xjl‖xjl ‖

form an orthonormal basis for Cn. For each j define the complex linear mapping

Λjl (xjl ) = ejl , l = 1, 2, . . . n.

The Pinchuk rescaling sequence associated to pj is then the sequence of maps

σj := Λj ψj : Cn → Cn.

Now assume that pj = φj(p0) for some p0 ∈ D and φj ∈ Aut(D). Let ωj = σj φj : D → Cn.One then has the following result [47]:

Proposition 2.5. (i) A subsequence of ωj converges uniformly on compact subsets to a holo-morphic embedding ω : D → Cn.

(ii) For the above subsequence ωj(D) converges, in the local Hausdorff topology, to ω(D).

(iii) Let q = limj→∞ pj be a smooth boundary point in the sense that there is a sphere contactfrom inside D at q. Then the 1-dimensional slices

Σj = z ∈ D : z − qj = λ(pj − qj)converge in the local Hausdorff topology to the upper-half plane (z1, 0, ..., 0) ∈ Cn : Re(z1) ≥ 0.In particular, the map z 7→ z + te1 is an automorphism of ω(D) for each t ∈ R.

3. Domains with positive-dimensional automorphism groups

In this section, we discuss the following basic question: Which Lie groups arise as automorphismgroups of bounded domains in Cn? There are several ways to approach this problem. Let us beginwith the Saerens–Zame [55] construction in which a given connected compact Lie group G isshown to arise as the full automorphism group of a smooth real-analytic strongly pseudoconvexdomain. This is done in two steps. In the first step, one constructs a domain on which Gacts while the second step consists of trimming away this domain to remove all extraneousautomorphisms, leaving G as the full automorphism group.

Note that G embeds in the complex unitary group U(n) for some large n > 0, which may be

viewed as a subgroup of GL(n,C) ' Cn2. Consider the action of G on GL(n,C) × Cm (where

m is chosen later) defined by

(g, (z, w)) 7→ (g · z, w)

for g ∈ G, z = (zij) ∈ GL(n,Cn) and w ∈ Cm. The real-valued function

φ(z, w) = | det(zij)|−2 +∑|zij |2 + |w|2


is a real-analytic strongly plurisubharmonic exhaustion for GL(n,C)×Cm. Now average out φover G to get

φG(z, w) =

∫Gφ(g · z, w) dg,

which is a real-analytic strongly plurisubharmonic function that is additionally G-invariant.Here dg is the normalised Haar measure on G. By Sard’s theorem, almost every t is such thatdφG 6= 0 on φG = t. Choose t0 that satisfies U(n)× 0 ⊂ φG < t0 in addition to dφG 6= 0on the level set φG = t0. The connected component D0 of φG = t0 that contains U(n)×0is a real-analytic strongly pseudoconvex domain on which G acts. By Fefferman’s theorem, eachelement of G (which is an automorphism of D0) extends smoothly up to the boundary ∂D0.What remains to be done is to ensure that there are no other automorphisms apart from thosein G. To this end, it will suffice to show that D0 can be perturbed slightly to get anotherG-invariant domain D such that ∂D is not spherical at any point and that if x, y are two pointson ∂D with x not in the G-orbit of y, then the curvature invariants of ∂D at x, y are distinct.This is where the role of transversality becomes evident and the choice of m becomes relevant.To clarify these steps further, let us briefly digress and recall the curvature invariants associatedwith nonspherical strongly pseudoconvex hypersurfaces that were introduced by Burns–Shnider–Wells [10]:

Let the variables in Cn+1 = Cn × C be (z, w) where z = (z1, z2, . . . , zn) and w = u+ iv. LetM ⊂ Cn+1 be a germ of a smooth strongly pseudoconvex hypersurface near the origin that isdefined by ψ = 0; here ψ is smooth and ψ(0) = 0. By diagonalizing the Levi form at the origin,ψ takes the form

ψ = v − 〈z, z〉 −N.Here 〈z, z〉 = |z1|2 + |z2|2 + . . .+ |zn|2 and

N =∑p,q≥2

Np,q

where Np,q is a polynomial of type (p, q) in z with coefficients that depend on u:

Np,q =∑

Nα1...αp,β1...βq(u)zα1

1 zα22 . . . z

αpp zβ11 z

β22 . . . z

βqq

and each coefficient Nα1...αp,β1...βq(u) can be expanded as a power series in u as

Nα1...αp,β1...βq(u) =

∑l≥0

N(l)

α1...αp,β1...βqul.

The origin is said to be a spherical point if∑∣∣∣N (0)

α1α2,β1β2

∣∣∣2 = 0;

else it is nonspherical. In the nonspherical case, a further change of coordinates of the formz∗ = ρz, w∗ = ρ2w, ρ > 0 can be made to ensure that this sum is 1. This is the restrictednormal form of ψ as defined in [10] and a pair of such representations for ψ differ only by a U(n)action in z. With ψ in such a form, the curvature invariants are defined for j ≥ 0, p ≥ q ≥ 2 by

K(l)p,q =

∑∣∣∣∣N (l)

α1...αp,β1...βq

∣∣∣∣2 .These are CR invariants associated to M at the origin, which is assumed to be a nonspherical

point. Note that K(0)2,2 = 1 by definition. Also, the scalars K

(l)p,q depend on the values of finitely

many derivatives of ψ at the origin and hence they may be regarded as functions on a suitablebundle of jets of real-valued functions in Cn+1 that vanish at the origin.


Returning back to the G-invariant domain D0, since G acts by left multiplication on the firstfactor, it follows that D0/G is a smooth real-analytic manifold that can be canonically identifiedwith an open subset of (GL(n,C)/G) × Cm. Let π : D0 → D0/G be the projection, which canbe seen to be proper since G is compact. Let C∞G (D0) and C∞(D0/G) be the spaces of smoothG-invariant functions on D0 and smooth functions on D0/G respectively. Likewise, let FG(D0)be the space of smooth G-invariant strongly plurisubharmonic functions with non-vanishingdifferential on D0 and let F(D0/G) consist of those smooth functions τ ∈ C∞(D0/G) such thatτ π ∈ F(D0). Note that φG ∈ F(D0). Since π is proper, the induced map

π∗ : C∞(D0/G)→ C∞G (D0)

is continuous and as F(D0) ⊂ C∞G (D0) is open in the relative topology, it follows that F(D0/G) =(π∗)−1(FG(D0)) is open in C∞(D0/G).

As is customary, let jkf(x) denote the k-jet of a smooth real-valued function f at a pointx in its domain of definition. Let J k(D0) be the jet bundle of order k of smooth real-valuedfunctions on D0 and let J kG(D0) ⊂ J k(D0) consist of k-jets of functions in FG(D0) at points in

D0. Likewise, let J k(D0/G) be the bundle of k-jets of functions in F(D0/G) at points of D0/G.The projection π naturally induces a smooth map (the push forward map)

πk∗ : J kG(D0)→ J k(D0/G)

which is defined as follows. Note that every jkφ(x) ∈ J kG(D0) is the k-jet of a smooth G-invariantstrongly plurisubharmonic function at some x ∈ D0. This φ descends to a well-defined function,say ψ on D0/G due to G-invariance. Evidently, φ = ψ π and hence

πk∗ (jkφ(x)) = jkψ(y) ∈ J k(D0/G)

is well-defined, where π(x) = y ∈ D0/G. It makes sense to speak of both spherical and non-spherical jets but the nonspherical ones are of interest here. Let SkG ⊂ J kG(D0) be the collectionof spherical jets. This is a locally closed real-analytic set since it is given by the vanishing of allthe curvature invariants, which may be regarded as functions on jet bundles. The other object ofinterest here is the locus of coinciding curvature invariants, which consists of all pairs of k-jets,say jkφ1(x), jkφ2(y), that have the same curvature invariants. Again, this is a locally closedreal-analytic set in the product J kG(D0) × J kG(D0). The main idea in [55] is to estimate thecodimensions of these real-analytic loci. It turns out that for a large enough m, the codimen-sions of these sets are large and hence by appealing to transversality arguments, it is possibleto find a φG, arbitrarily close to φG on D0, whose level surface is not spherical anywhere. LetD = φG < t0. Then D is bounded, ∂D is nonspherical everywhere and each pair x, y ∈ ∂Dwith x not in the G-orbit of y have distinct curvature invariants.

It remains to show that Aut(D) = G, i.e., for each h ∈ Aut(D), there is a g ∈ G such thath(z, w) = (g, (z, w)). To do this, the first step is to show that for a given h ∈ Aut(D) and(z, w) ∈ D, there exists a unique g = g(z, w) ∈ G such that h(z, w) = (g, (z, w)). Granting this,note then that

h(z, w) = (g(z, w) · z, w) = (h1(z, w), h2(z, w))

where h1 : D → GL(n,C) is holomorphic. But then g(z, w) = h1(z, w) · z−1 is a holomorphicmap from D with values in G ⊂ U(n). Since U(n) ⊂ GL(n,C) is maximally totally real, itfollows that g(z, w) must be constant and hence h ∈ Aut(D) can be identified with g ∈ G. Thiscompletes the proof.

Winkelmann extended this by showing that for every connected real Lie group G, thereexists a Stein complete Kobayashi hyperbolic manifold X on which G acts effectively, freely andproperly with totally real orbits such that G ' Aut(X). There are two essential difficulties thatneed to be overcome in following the Saerens–Zame approach. First, not every noncompact Lie


group is linear and secondly, the Wong–Rosay theorem (that will be discussed later) implies thata smooth strongly pseudoconvex domain with noncompact automorphism group is necessarilybiholomorphic to the ball. Thus, in general, the construction of a suitable domain on which thegiven noncompact G acts will either be unbounded or necessarily have nonsmooth boundarysomewhere. Therefore it is no longer evident that for each automorphism of this domain, thereis a sequence pn that converges to a smooth boundary point in such a way that φ(pn) alsoconverges to a smooth boundary point. Having such data is essential to show that φ extendssmoothly to the smooth part of the boundary. The first problem is resolved by using Ado’stheorem according to which every Lie algebra is linear and working with coverings to reduceto the case when G is simply connected. The second problem is handled in two steps. First,Privalov’s theorem is used to ensure that the cluster set of a smooth boundary point intersectsa possibly different smooth portion of the boundary. Once this is done, the second step is to usea local regularity theorem for continuous CR homeomorphisms between smooth real-analyticstrongly pseudoconvex hypersurfaces – for example, by Bell.

A related theorem of Shabat–Tumanov [54] shows that every connected linear Lie group arisesas the full automorphism group of a strongly pseudoconvex domain, which is possibly unboundedbut is biholomorphic to a bounded domain.

In the Bedford–Dadok approach [2], the domain on which G acts sits inside the complexifi-cation GC (or at most GC × C). The domain in fact can be taken to be a small G-invarianttubular neighbourhood of G in GC. The fact that such a domain does not have any other au-tomorphisms other than those in G uses a theorem of Ochiai–Takahashi, which says that anyisometry of a compact, connected, simple Lie group G is a composition of a left translation witha right translation.

Another approach, due to S. J. Kan [40], uses the canonical complexification of a real-analyticRiemannian manifold by the Grauert tube construction. Let (M, g) be a real-analytic Rie-mannian manifold of dimension at least 2. Identify M with the zero section in TM and letρ(x, v) = |v|2, where v ∈ TxM , be the length of v with respect to g. Then there exists a com-plex structure on all sufficiently small neighbourhoods of M in TM with respect to which ρ isstrongly plurisubharmonic and satisfies (ddcρ)n = 0 away from M . The Grauert tube of radiusr > 0 and center M is

T rM = (x, v) : x ∈M, v ∈ TxM, |v| < r = ρ−1([0, r)),

and this is Stein for small r if M is compact in particular. Thus M = ρ = 0 being the zero setof a non-negative strongly plurisubharmonic function can be viewed as a totally real submanifoldof TM . This construction can be applied to the compact Lie group G as well and the domainD on which G acts turns out to be the sub-level set of a suitable perturbation of ρ. By showingthat D is not biholomorphic to the ball, it follows that Aut(D) ⊂ Isom(G) ' L(G) · K (hereL(G) are the left translations of G and K is the isotropy subgroup). The final step here is toshow that K ' Id and that Aut(D) ' L(G) ' G, which completes the proof.

Whether an arbitrary real Lie group can be realized as the full automorphism group of adomain seems to be open. However, here is an indication that this might be true. In [66],Winkelmann showed that every countable group can be realized as the full automorphism groupof a Riemann surface as well as the full group of isometries of a Riemannian manifold.

4. Miscellaneous Topics

4.1. Variation of the automorphism group with D. It is of interest to study how Aut(D)varies with D. For this, a suitable topology on the space of bounded domains is required and


there are several to choose from. To give a flavour of what’s known about this question, we willdiscuss the following theorem:

Greene–Krantz [28], [29] considered C∞-smooth perturbations of a given strongly pseudo-convex domain D0 ⊂ Cn. For all domains D that are C∞ close to D0, there exists a C∞

diffeomorphism f : D → D0 such that for every g ∈ Aut(D), the map f g f−1 ∈ Aut(D0). Inparticular, this says that Aut(D) is isomorphic to a subgroup of Aut(D0). This can be viewedas a statement about the upper semicontinuity of Aut(D) as a function of D. There are twocases to consider.

When D0 ' Bn, there is a C∞ neighbourhood U of D0 with the property that if D ∈ U , thenD satisfies the following properties: One, it is a smooth strongly pseudoconvex domain (sincethis is clearly an open condition) and second, the Bergman metric of D has negative Riemanniansectional curvature everywhere. The latter property holds since the Bergman kernel and hencethe associated metric is stable under smooth perturbations – see [30]. If D ' Bn, then theproof is complete since any biholomorphism f : D → D0 w Bn can be used as a conjugatingdiffeomorphism as required in the theorem. When D 6w Bn, the Wong–Rosay theorem impliesthat Aut(D) is compact. Using the fact that Aut(D) acts on D isometrically in the Bergmanmetric, it follows that Aut(D) has a fixed point in D, say p. Furthermore, D is simply connectedsince it is a small perturbation of the ball, and the Bergman metric on it is complete and isnegatively curved. By the Cartan–Hadamard theorem, the exponential map exp : TpD → D is adiffeomorphism. The exponential map conjugates the action of Aut(D) on D to multiplicationby the derivatives of the automorphisms, which is a unitary map. This is conjugate to a subgroupof the unitary group acting on the ball via a diffeomorphism and this completes the proof inthis case.

The case when D0 6' Bn uses Ebin’s theorem in the Riemannian case. Let us recall it for thesake of completeness: Suppose M is a smooth compact Riemannian manifold. Let γj (j ≥ 1),γ be smooth Riemannian metrics on M such that γj → γ in the C∞ topology. Assume thatthe isometry group Iso(γ) with respect to γ is compact. Then the isometry group Iso(γj) isLie isomorphic to a subgroup of Iso(γ) for all large j. Furthermore, the Lie isomorphism canbe realized as a conjugation by a diffeomorphism of M . Note that Aut(D0) is compact. Thereis, in this case, a smooth metric on D0 that is Aut(D0) invariant and whose isometry groupcoincides with that of the Bergman metric. The construction of such a metric is stable undersmooth perturbations and thus all nearby domains carry such a metric. Let D be near D0 andfix a diffeomorphism F : D → D0 that is close to the identity. Also, let γ be such a metricon D0. Then F∗γ is another such metric on D0 that is close to γ. The isometry group of γis compact being the union of Aut(D0) and the holomorphic conjugates of elements therein.By considering the double of D0, which is compact, and by Ebin’s theorem, it follows that theisometry groups of F∗γ are isomorphic to subgroups of the isometry groups of γ. In particular,what this says is that for all elements of the form Fα = F α F−1, α ∈ Aut(D), there is asmooth diffeomorphism f of the double of D0, close to the identity, such that f Fα f−1 is anisometry of γ on the double of D0. As noted above, all such elemets are either holomorphic orconjugate holomorphic. The latter is ruled out since f, F are close to the identity. It remains toshow that f , which is defined on the double of D0, preserves D0. Thus (f F )−1 is the requireddiffeomorphism.

Recent work has focussed on similar semicontinuity results for finitely smooth strongly pseu-doconvex domains – see for example [27].

It is also interesting to consider the map D 7→ dim Aut(D) where the space of bounded do-mains is equipped with the Hausdorff distance between their boundaries. Fridman–Ma–Poletsky


[20] showed that this map is upper semicontinuous and hence a given domain cannot be approx-imated in the Hausdorff metric by domains whose automorphism groups have strictly largerdimensions.

4.2. Determining sets for automorphisms. For a complex manifold M , a subset K ⊂ Mis a determining set if each automorphism g of M that satisfies g(x) = x for every x ∈ K is theidentity map on M . This definition is motivated from the planar case wherein three distinctpoints suffice. Work of Fridman–Kim–Krantz–Ma [18], [19] shows that:

(i) for every finite K ⊂ Cn, n ≥ 2, there exists a bounded domain D containing K and asubgroup H ⊂ Aut(D) isomorphic to the unitary group U(n− 1) such that each h ∈ Hfixes K pointwise,

(ii) for a bounded pseudoconvex domain D ⊂ Cn, almost every n-tuple of points is a deter-mining set for D, and

(iii) determining sets are preserved under small perturbations of the domain.

Thus, (i) says that most finite sets are not determining sets if we choose D appropriately, and onthe other hand (ii) says that for a fixed D, most n-tuples are indeed determining sets. To see howto prove (i), let K = p1, p2, . . . , pk ⊂ Cn be a given finite set. Write pj = (uj , vj) ∈ C×Cn−1.By a rotation of coordinates followed by a scaling, we may assume that the uj are all distinctand that |uj | < 1 for all j. Let f : C→ Cn−1 be a holomorphic map that satisfies f(uj) = vj forall j. For example, the Lagrange interpolation polynomial that maps uj to the l-th coordinateof vj can be taken to be the l-th component of f . The map F (z) = w where

w1 = z1, w′ = z′ + f(z1)

is an automorphism of Cn that satisfies F (uj , 0) = pj . Thus, F moves the given set of points tothe z1-axis. Let D = F (Bn) and let U(n− 1) be the unitary subgroup acting in the z′ variables.Then H = F U(n− 1) F−1 is a subgroup of Aut(D) that fixes K pointwise.

The claim made in (ii) holds in greater generality than mentioned. To describe the main ideasthere, let M be a connected, complete Riemannian manifold. For x ∈ M , a point y ∈ M is acut point of x if there are at least two length minimizing geodesics from x to y. The collectionof all cut points of x will be denoted by Cx and it is known that it is a nowhere dense set inM . A set X ⊂M is a Cartan–Hadamard set if there exists an x0 ∈ X such that X ⊂M \ Cx0 .In this case, x0 will be called a pole of X and the pair will be denoted as (X,x0). Call sucha pair (X,x0) spanning if the following property is satisfied: for y ∈ X \ x0, consider thegeodesic γ from x0 to y. The derivative γ′(0) ∈ Tx0M . The collection of such tangent vectorsγ′(0) : y ∈ X \ x0 spans Tx0M .

The relevance of such spanning sets stems from Lemma 1.1 in [19]: If (X,x0) is a spanningCartan–Hadamard set and f : M → M is an isometry that fixes X pointwise, then f mustbe the identity on M . Indeed, the differential of f will have to fix each of the tangent vectorsγ′(0) since f is an isometry. As this collection of vectors spans Tx0M , it must follow that thedifferential of f , as a self map of Tx0M , must be the identity. Hence f fixes every point inM \ Cx0 and therefore every point in M .

Finding such spanning Cartan–Hadamard sets involves an inductive procedure that at eachstage picks a point that avoids the cut locus of all the previously defined points. Thus, if thedimension of M is n, then almost every collection of n+ 1 points defines such a spanning set. Itremains to run through these arguments in the case of bounded pseudoconvex domains, equippedwith the canonical Kahler–Einstein metric that is invariant under automorphisms.

The concluding claim in (iii) relies on normal family arguments and the semicontinuity ofautomorphism groups discussed above.


Related work of Fridman–Ma–Vigue [21], Vigue [63] extends this notion for holomorphicself-maps of manifolds where the role of hyperbolicity becomes relevant.

4.3. A characterization of Cn. For a domain D ⊂ Cn, Rubel raised the question of studying amore general object, namely, End(D), the semigroup of holomorphic endomorphisms of D. Thisis evidently a much larger object than Aut(D) and he asked whether End(D) determines D. A.Eremenko [15] showed that this is indeed so for a class of Riemann surfaces. More precisely,let X,Y be a pair of Riemann surfaces that admit nonconstant bounded holomorphic functionsand suppose that End(X) ' End(Y ). Then X and Y are either biholomorphic or conjugateholomorphic. This was extended for bounded domains in Cn by S. Merenkov [50] and this wasshown to hold for Cn as well by Buzzard–Merenkov [11] . To state this more precisely, let ψbe a bijection of Cn, n ≥ 2, onto a complex manifold M such that ψ f ψ−1 ∈ End(M) forevery f ∈ End(Cn). Then ψ is either biholomorphic or conjugate biholomorphic. The proofhas a dynamical flavour to it; the first step, which shows that such a ψ must necessarily bea homeomorphism, involves working with Fatou–Bieberbach domains that arise as basins ofattraction of Henon maps and omit a codimension 2 analytic set in Cn.

Related work of Isaev [35] shows that if M is Stein and Aut(M) ' Aut(Cn), then M isbiholomorphic to Cn. The idea here is to linearize the torus action Tn, which shows that Mmust be biholomorphic to either Cn or Cn minus some hyperplanes. The latter has disconnectedautomorphism group and this leaves only the first possibility.

It must also be mentioned that Isaev–Kruzhilin [39] have classified connected n-dimensionalcomplex manifolds that admit effective actions of the unitary group U(n) by biholomorphictransformations. Their proof relies on first identifying all possible orbits that can occur. Itturns out that an orbit can either be a point (in which case U(n) has a fixed point), a realhypersurface, a complex hypersurface or the whole of M , i.e., M is homogeneous. When thereis a fixed point, results of Kaup [41] show that M is biholomorphic to either Bn, Cn or CPn.When the orbits are hypersurfaces (either real or complex), the following dichotomy is studied:

(i) All orbits are real hypersurfaces: In this case, M must either be a spherical shell in Cn,or a Hopf manifold, or a quotient of one of these by a suitable discrete subgroup.

(ii) There exists at least one orbit that is a complex hypersurface: In this case, there canbe at most two orbits that are complex hypersurfaces and these turn out to be copiesof CPn−1. This is possible only when M is obtained by blowing up either Bn or Cn atthe origin, or attaching the hyperplane at infinity to the exterior of the unit ball, orblowing up CPn at a point, or by taking a quotient of one of these by a suitable discretesubgroup.

In the homogeneous case, M turns out to be the quotient of a Hopf manifold by a suitablediscrete subgroup. The following characterization of Cn is a consequence of this classificationtheorem:

If M is a connected complex manifold of dimension n such that Aut(M) and Aut(Cn) areisomorphic as topological groups, then M is biholomorphic to Cn.

Note that M is not assumed to be Stein in this theorem.

4.4. Automorphisms of complements of analytic sets. To begin with, note that Cartan’stheorem about Aut(D) being a finite-dimensional real Lie group requires D to be bounded.Indeed, Kaup [41] described a general construction of unbounded domains Ω ⊂ Cn such thatAut(Ω) acts transitively on Ω but nevertheless no finite-dimensional Lie group acts transitivelyon it. A special case of this can be described as follows. Let

Ω = (n, 0) ∈ C2 : n ∈ N


and let h ∈ O(C) be such that h(n) = 0 for all n ≥ 1. Then Aut(Ω) contains maps of the form

f(z1, z2) = (z1, z2 + h(z1)) and g(z1, z2) = (z1 + z2, z2).

The collection of such maps is seen to act transitively on Ω since f moves points vertically keepingthe first coordinate intact while g moves points in the horizontal direction. If Ω were to admita transitive G action for some real Lie group G, then Ω ' G/I for some closed subgroup I andhence all higher homotopy groups πk(Ω), k > 1 would have to be finitely generated. However,π3(Ω) is not finitely generated. Hence no finite-dimensional Lie group can act transitively on it.Note that Ω is not Stein.

Winkelmann studied the properties of complements of analytic sets in Cn in [65] and inparticular showed that X = C2 \ A where A = (z1, z2) ∈ C2 : z1z2 = 1, which is Stein, hassuch a property. That is, Aut(X) acts transitively on X but it does not admit a transitiveG-action for any finite-dimensional real or complex Lie group G. Continuing with this theme,a sufficient condition for a domain D to admit a transitive Aut(D) is given. To describe thiscriterion, let Y be a complex manifold equipped with the Kobayashi metric dkY . For y ∈ Y , let

Sy = x ∈ Y \ y : dkY (x, y) = 0 be the degeneracy locus of y and define EY ⊂ Y to be thesmallest analytic subset that contains all those y ∈ Y for which Sy is nonempty. Call Y almosthyperbolic if EY 6= Y . Note that if EY = ∅ then Y is (Kobayashi) hyperbolic. A nonconstantholomorphic map f from a complex manifold X to an almost hyperbolic Y is nondegenerate iff(X) 6⊂ EY . It is shown that if X = Cn \ A, where A ⊂ Cn is analytic, admits a transitiveAut(X) action, then there does not exist a nondegenerate holomorphic map from X to anyalmost hyperbolic manifold Y . In particular, let f ∈ O(Cn) be a nonconstant holomorphic mapand let X = Cn \ (f = 0 ∪ f = 1). Then Aut(X) does not act transitively on it.

It seems appropriate to conclude by reporting on some recent results in these areas. Here aretwo theorems of Zimmer [68], [69]:

(a) A bounded generic analytic polyhedron in Cn with noncompact automorphism group isbiholomorphic to ∆r ×W for some r > 0 where W is a complex manifold with Aut(W )compact. Note that there is no convexity hypothesis on the polyhedron. This extendsthe work of Kim and Kim–Krantz–Spiro mentioned earlier.

(b) A smoothly bounded convex domain D ⊂ Cn is biholomorphic to a weighted homo-geneous polynomial domain if and only if the set of orbit accumulation points on ∂Dintersects at least two closed complex faces. Here, a closed complex face by definitionmeans the intersection of the complex tangent spaces to points on ∂D with ∂D. In otherwords, a face at p ∈ ∂D is the set TC

p (∂D) ∩ ∂D. The domain D is not assumed to beof finite type and the key step is to establish a version of the Greene–Krantz conjecturefor smoothly bounded convex domains. In short, this theorem can be interpreted asa characterization of smooth convex domains that have a ‘larger’ automorphism groupthan merely being noncompact.

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[69] A. Zimmer: Characterizing polynomial domains by their automorphism group, preprint avaliable at:https://arxiv.org/pdf/1506.07852.pdf

Index

Bedford-Dadok construction, 16Bedford-Pinchuk theorem, 9

Cartan’s principle, 2

determining set, 18

Frankel’s theorem, 11

Greene-Krantz conjecture, 3

Kan’s construction, 16Kim-Krantz-Pinchuk rescaling, 12

Pinchuk rescaling, 9proper action, 1

Saerens-Zame construction, 13Siegel’s theorem, 4

Winkelmann’s theorem, 15Wong-Rosay theorem, 8

24


Harish Seshadri: Department of Mathematics, Indian Institute of Science, Bangalore 560 012,India

E-mail address: [email protected]

Kaushal Verma: Department of Mathematics, Indian Institute of Science, Bangalore 560 012,India

E-mail address: [email protected]

contentsharish/papers/auto-survey.pdf · harish seshadri, kaushal verma contents 1. introduction 1...

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