Characterizing strong pseudoconvexity, obstructions to biholomorphisms, and Lyapunov exponents
aa r X i v : . [ m a t h . C V ] A p r CHARACTERIZING STRONG PSEUDOCONVEXITY,OBSTRUCTIONS TO BIHOLOMORPHISMS, AND LYAPUNOVEXPONENTS
ANDREW ZIMMER
Abstract.
In this paper we consider the following question: For boundeddomains with smooth boundary, can strong pseudoconvexity be characterizedin terms of the intrinsic complex geometry of the domain? Our approach toanswering this question is based on understanding the dynamical behavior ofreal geodesics in the Kobayashi metric and allows us to prove a number ofresults for domains with low regularity. For instance, we show that for convexdomains with C ,ǫ boundary strong pseudoconvexity can be characterized interms of the behavior of the squeezing function near the boundary, the be-havior of the holomorphic sectional curvature of the Bergman metric near theboundary, or any other reasonable measure of the complex geometry near theboundary. The first characterization gives a partial answer to a question ofFornæss and Wold. As an application of these characterizations, we show thata convex domain with C ,ǫ boundary which is biholomorphic to a stronglypseudoconvex domain is also strongly pseudoconvex. Introduction
A domain in C d with C boundary is called strongly pseudoconvex if the Leviform of the boundary is positive definite. The Levi form is extrinsic and in thispaper we study the following question: Question . For domains with C boundary, can strong pseudoconvexity be char-acterized in terms of the intrinsic complex geometry of the domain?Although strongly pseudoconvex domains form one of the most important classesof domains in several complex variables, it does not appear that Question 1 has beenextensively studied. The only general results we know of are due to Bland [Bla85,Bla89], who studies compactifications of complete simply connected non-positivelycurved K¨ahler manifolds whose curvature tensor approaches the curvature tensor ofcomplex hyperbolic space in a controlled way. Under these conditions, Bland provesthat the geodesic compactification has a natural CR-structure which is stronglypseudoconvex and uses this to construct bounded holomorphic functions.In this paper we will consider only domains in C d , but will avoid needing tocontrol how fast the geometry of the domain approaches the geometry of complexhyperbolic space. We will also focus on the case of convex domains. Convexity isa strong geometric assumption, but in relation to Bland’s results can be seen as anon-positive curvature condition. By assuming convexity we are also able to proveresults about unbounded domains and domains whose boundary has low regularity. Department of Mathematics, College of William and Mary, Williamsburg, VA 23185
E-mail address : [email protected] . Date : April 20, 2018.
Our approach to studying Question 1 is based on understanding the behavior ofthe real geodesics in the Kobayashi metric. Let B d ⊂ C d denote the open unit balland K B d denote the Kobayashi distance on B d . Then geodesics in ( B d , K B d ) havethe following properties:(1) if γ , γ : R ≥ → B d are geodesics and lim inf s,t →∞ K B d ( γ ( s ) , γ ( t )) < ∞ ,then there exists T ∈ R such that lim t →∞ K B d ( γ ( t ) , γ ( t + T )) = 0 and(2) if γ , γ : R ≥ → B d are geodesics and lim t →∞ K B d ( γ ( t ) , γ ( t )) = 0, thenlim t →∞ t log K B d ( γ ( t ) , γ ( t )) = − γ , γ are contained in the same complex geodesic andlim t →∞ t log K B d ( γ ( t ) , γ ( t )) = − ± ± Domains biholomorphic to strongly pseudoconvex domains.
One ofour motivations for studying Question 1 is the following question of Fornæss andWold.
Question . (Fornæss and Wold [FW16, Question 4.5]) Suppose Ω ⊂ C d is abounded domain with C boundary and Ω is biholomorphic to the unit ball in C d . Is Ω strongly pseudoconvex?One can also ask the following more general question: Question . Suppose Ω , Ω ⊂ C d are bounded domains with C boundary, Ω is strongly pseudoconvex, and Ω is biholomorphic to Ω . Is Ω also stronglypseudoconvex?When Ω and Ω both have C ∞ boundary, Bell [Bel81] answered the abovequestion in the affirmative using deep analytic methods, namely condition (R) andKohn’s subelliptic estimates in weighted L -spaces. It does not appear that Bell’sanalytic approach can be used in the C regularity case.Using the dynamical approach described above, we will establish the followingpartial answer to Question 3. Theorem 1.1.
Suppose Ω ⊂ C d is a bounded strongly pseudoconvex domain with C boundary and C ⊂ C d is a convex domain biholomorphic to Ω . If C has C ,α boundary for some α > , then every x ∈ ∂ C is a strongly pseudoconvex point of ∂ C .Remark . Theorem 1.1 makes no assumptions about the boundedness of C .The dynamical approach also allows us to prove a theorem for convex domainswith only C boundary, but we need to introduce some additional notation. BSTRUCTIONS TO BIHOLOMORPHISMS 3
Definition 1.3.
For a domain Ω ⊂ C d , a point z ∈ Ω, and a non-zero vector v ∈ C d define δ Ω ( z ) = inf {k z − w k : w ∈ ∂ Ω } and δ Ω ( z ; v ) = inf {k z − w k : w ∈ ∂ Ω ∩ ( z + C · v ) } . We will then prove the following.
Theorem 1.4. (see Section 4) Suppose Ω ⊂ C d is a bounded strongly pseudoconvexdomain with C boundary and C ⊂ C d is a convex domain biholomorphic to Ω . If C has C boundary, then for every ǫ > and R > there exists a C = C ( ǫ, R ) ≥ such that δ C ( z ; v ) ≤ Cδ C ( z ) / (2+ ǫ ) for all z ∈ C with k z k ≤ R and all nonzero v ∈ C d .Remark . (1) Suppose Ω ⊂ C d is bounded, convex, and has C boundary. Then Ω isstrongly pseudoconvex if and only if there exists a C ≥ δ Ω ( z ; v ) ≤ Cδ Ω ( z ) / for all z ∈ Ω and all nonzero v ∈ C d . Thus the conclusion of Theorem 1.4can be interpreted as saying C is “almost” strongly pseudoconvex.(2) By picking ǫ < α , one sees that Theorem 1.1 is a corollary of Theorem 1.4.1.2. The intrinsic complex geometry of a domain.
There are many ways tomeasure the complex geometry of a domain and in this subsection we describehow certain natural measures provide characterizations of strong pseudoconvexityamongst convex domains with C ,α boundary. As we will describe in Subsection 5.1,a recent example of Fornæss and Wold [FW16] shows that all these characterizationsfail for convex domains with C boundary.1.2.1. The squeezing function.
One natural intrinsic measure of the complex geom-etry of a domain is the squeezing function. Given a bounded domain Ω ⊂ C d let s Ω : Ω → (0 ,
1] be the squeezing function on
Ω, that is s Ω ( p ) = sup { r : there exists an one-to-one holomorphic map f : Ω → B d with f ( p ) = 0 and r B d ⊂ f (Ω) } . Although only recently introduced, the squeezing function has a number of appli-cations, see for instance [LSY04, Yeu09].Work of Diederich, Fornæss, and Wold [DFW14, Theorem 1.1] and Deng, Guan,and Zhang [DGZ16, Theorem 1.1] implies the following theorem.
Theorem 1.6. [DFW14, DGZ16] If Ω ⊂ C d is a bounded strongly pseudoconvexdomain with C boundary, then lim z → ∂ Ω s Ω ( z ) = 1 . Based on the above theorem, it seems natural to ask if the converse holds.
OBSTRUCTIONS TO BIHOLOMORPHISMS
Question . (Fornæss and Wold [FW16, Question 4.2]) Suppose Ω ⊂ C d is abounded pseudoconvex domain with C k boundary for some k >
2. Iflim z → ∂ Ω s Ω ( z ) = 1 , is Ω strongly pseudoconvex?Surprisingly the answer is no when k = 2: Fornæss and Wold [FW16] constructeda convex domain with C boundary which is not strongly pseudoconvex, but thesqueezing function still approaches one on the boundary. However, we will provethat a little bit more regularity is enough for an affirmative answer. Theorem 1.7. (see Subsection 5.5) For any d ≥ and α > , there exists some ǫ = ǫ ( d, α ) > such that: if Ω ⊂ C d is a bounded convex domain with C ,α boundaryand s Ω ( z ) ≥ − ǫ outside a compact subset of Ω , then Ω is strongly pseudoconvex.Remark . Using a different argument, we previously gave an affirmative answerto Question 4 for bounded convex domains with C ∞ boundary [Zim16]. Moreover,Joo and Kim [JK16] gave an affirmative answer for bounded finite type domains in C with C ∞ boundary.1.2.2. Holomorphic sectional curvature of the Bergman metric.
Another intrinsicmeasure of the complex geometry of a domain is the curvature of the Bergmanmetric.Let (
X, J ) be a complex manifold with K¨ahler metric g . If R is the Riemanniancurvature tensor of ( X, g ), then the holomorphic sectional curvature H g ( v ) of anonzero vector v is defined to be the sectional curvature of the 2-plane spanned by v and Jv , that is H g ( v ) := R ( v, Jv, Jv, v ) k v k g . A classical result of Hawley [Haw53] and Igusa [Igu54] says that if (
X, g ) is acomplete simply connected K¨ahler manifold with constant negative holomorphicsectional curvature, then X is biholomorphic to the unit ball (also see ChapterIX, Section 7 in [KN96]). Moreover, if b B d is the Bergman metric on the unit ball B d ⊂ C d , then ( B d , b B d ) has constant holomorphic sectional curvature − / ( d + 1).Klembeck proved that the holomorphic sectional curvature of Bergman metric ona strongly pseudoconvex domain approaches − / ( d + 1) on the boundary. Theorem 1.9 (Klembeck [Kle78]) . Suppose Ω ⊂ C d is a bounded strongly pseudo-convex domain with C ∞ boundary. Then lim z → ∂ Ω max v ∈ T z Ω \{ } (cid:12)(cid:12)(cid:12)(cid:12) H b Ω ( v ) − − d + 1 (cid:12)(cid:12)(cid:12)(cid:12) = 0 where b Ω is the Bergman metric on Ω . We will prove the following converse to Klembeck’s theorem:
BSTRUCTIONS TO BIHOLOMORPHISMS 5
Theorem 1.10. (see Subsection 5.7) For any d ≥ and α > , there exists some ǫ = ǫ ( d, α ) > such that: if Ω ⊂ C d is a bounded convex domain with C ,α boundaryand max v ∈ T z Ω \{ } (cid:12)(cid:12)(cid:12)(cid:12) H b Ω ( v ) − − d + 1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ outside a compact subset of Ω , then Ω is strongly pseudoconvex. K¨ahler metrics with controlled geometry.
In Subsection 5.6 we will introducefamilies of K¨ahler metrics, denoted by G M (Ω) for some M >
1, on a convex domainΩ which have controlled geometry relative to the Kobayashi metric. We will alsoshow that there exists some M > G M (Ω) when M ≥ M . Then we will prove the following generalizationof Theorem 1.10. Theorem 1.11. (see Subsection 5.6) For any d ≥ , α > , and M > , thereexists some ǫ = ǫ ( d, α, M ) > such that: if Ω ⊂ C d is a bounded convex domainwith C ,α boundary and there exists a metric g ∈ G M (Ω) with max v,w ∈ T z Ω \{ } | H g ( v ) − H g ( w ) | ≤ ǫ outside a compact subset of Ω , then Ω is strongly pseudoconvex. Other intrinsic measures of the complex geometry of a domain.
Theorem 1.7,Theorem 1.10, and Theorem 1.11 are particular cases of more general theoremswhich we state and prove in Section 5. These more general theorems extend The-orem 1.7, Theorem 1.10, and Theorem 1.11 to essentially any intrinsic measure ofthe complex geometry of a domain.1.3.
Some notations. (1) For z ∈ C d , let k z k denote the standard Euclidean norm.(2) For a point z ∈ C d and r >
0, let B d ( z ; r ) = { w ∈ C d : k w − z k < r } . (3) D ⊂ C will denote the open unit disk and B d := B d (0; 1) ⊂ C d will denotethe open unit ball.(4) Let D = { z ∈ C : | Im( z ) | + | Re( z ) | < } . (5) If C ⊂ C d is a convex domain with C boundary and ξ ∈ ∂ C let T C ξ ∂ C ⊂ C d denote the complex tangent space of ∂ C at ξ . Then since C is convex andopen (cid:0) ξ + T C ξ ∂ C (cid:1) ∩ C = ∅ . Acknowledgments.
I would like to thank the referee for a number of commentsand corrections which improved the present work. This material is based uponwork supported by the National Science Foundation under grants DMS-1400919and DMS-1760233.
OBSTRUCTIONS TO BIHOLOMORPHISMS Lyapunov exponents and the shape of the boundary
In this section we establish a relationship between the “Lyapunov exponents ofthe geodesic flow” and the shape of the boundary. This relationship allows us toprove the following result.
Proposition 2.1.
Suppose d ≥ and C ⊂ C d is a convex domain with the followingproperties: (1) C ∩
Span C { e , . . . , e d } = ∅ , (2) C ∩ C · e = { ze : Im( z ) > } , and (3) C is biholomorphic to the unit ball.Then lim r →∞ r log δ C ( ie r e ; v ) = 1 / for all v ∈ Span C { e , . . . , e d } .Remark . The unit ball is biholomorphic to the convex domain P d = ( ( z , . . . , z d ) ∈ C d : Im( z ) > d X i =2 | z i | ) and this domain satisfies:(a) P d ∩ Span C { e , . . . , e d } = ∅ ,(b) P d ∩ C · e = { ze : Im( z ) > } , and(c) δ P d ( ie r e ; v ) = e r/ for all r ∈ R and v ∈ Span C { e , . . . , e d } .Hence the above proposition states that if a convex domain is biholomorphic tothe unit ball and satisfies conditions (a) and (b) above, then the convex domainasymptotically satisfies condition (c).Before starting the proof of Proposition 2.1 we will recall some facts about theKobayashi pseudo-metric on convex domains and geodesics in complex hyperbolicspace.2.1. The Kobayashi metric and distance.
In this subsection we recall the defi-nition of the Kobayashi pseudo-metric. A more thorough introduction can be foundin [Kob05].Given a domain Ω ⊂ C d the (infinitesimal) Kobayashi pseudo-metric on Ω is thepseudo-Finsler metric k Ω ( x ; v ) = inf {| ξ | : f ∈ Hol(∆ , Ω) , f (0) = x, d ( f ) ( ξ ) = v } . Royden [Roy71, Proposition 3] proved that the Kobayashi pseudo-metric is an uppersemicontinuous function on Ω × C d . So, if σ : [ a, b ] → Ω is an absolutely continuouscurve (as a map [ a, b ] → C d ), then the function t ∈ [ a, b ] → k Ω ( σ ( t ); σ ′ ( t ))is integrable and we can define the length of σ to be ℓ Ω ( σ ) = Z ba k Ω ( σ ( t ); σ ′ ( t )) dt. BSTRUCTIONS TO BIHOLOMORPHISMS 7
One can then define the
Kobayashi pseudo-distance to be K Ω ( x, y ) = inf { ℓ Ω ( σ ) : σ : [ a, b ] → Ω is absolutely continuous , with σ ( a ) = x, and σ ( b ) = y } . This definition is equivalent to the standard definition of K Ω via analytic chains,see [Ven89, Theorem 3.1].Directly from the definition one obtains the following property of the Kobayashipseudo-metric: Proposition 2.3.
Suppose Ω ⊂ C d and Ω ⊂ C d are domains. If f : Ω → Ω is a holomorphic map, then K Ω ( f ( z ) , f ( w )) ≤ K Ω ( z, w ) for all z, w ∈ Ω . For a general domain Ω it is very hard to determine if (Ω , K Ω ) is a Cauchycomplete metric space, but for convex domains there is a very simple (to state)characterization due to Barth. Theorem 2.4 (Barth [Bar80, Theorem 1]) . Suppose Ω ⊂ C d is a convex domain.Then the following are equivalent: (1) Ω does not contain any complex affine lines, (2) K Ω is non-degenerate and hence a distance on Ω , (3) K Ω is a proper Cauchy complete distance on Ω ,Remark . To be precise, Theorem 1 in [Bar80] only states that conditions (1)and (2) are equivalent to K Ω being a proper distance on Ω. However, for lengthspaces any proper distance is also Cauchy complete, see for instance Corollary 3.8in [BH99, Chapter I].2.2. Basic estimates for the Kobayashi metric.
In this subsection we recallsome basic estimates for the Kobayashi metric on convex domains. All these esti-mates are very well known, but we provide the short proofs for the reader’s conve-nience.
Lemma 2.6.
Suppose Ω ⊂ C d is a convex domain, V ⊂ C d is a complex affineline, and V ∩ Ω is a half plane in V . Then K Ω ( z , z ) = K V ∩ Ω ( z , z ) for all z , z ∈ V ∩ Ω .Proof. By applying an affine transformation we may assume that(1) V ∩ Ω = { ( z, , . . . ,
0) : Im( z ) > } and(2) Ω ⊂ { ( z , . . . , z d ) : Im( z ) > } .Applying the distance decreasing property of the Kobayashi metric to the inclu-sion map V ∩ Ω ֒ → Ω implies that K Ω ( z , z ) ≤ K V ∩ Ω ( z , z )for all z , z ∈ V ∩ Ω. OBSTRUCTIONS TO BIHOLOMORPHISMS
Let P : C d → V denote the map P ( z , . . . , z d ) = ( z , , . . . , P (Ω) =Ω ∩ V and P ( z ) = z for z ∈ V . So applying the distance decreasing property of theKobayashi metric to the projection map P : Ω → V ∩ Ω implies that K V ∩ Ω ( z , z ) ≤ K Ω ( z , z )for all z , z ∈ V ∩ Ω. (cid:3) Lemma 2.7.
Suppose Ω ⊂ C d is a convex domain, H ⊂ C d is a complex affinehyperplane such that H ∩ Ω = ∅ , and P : C d → C is an affine map with P − (0) = H .Then for any z , z ∈ Ω we have K Ω ( z , z ) ≥ (cid:12)(cid:12)(cid:12)(cid:12) log (cid:12)(cid:12)(cid:12)(cid:12) P ( z ) P ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Proof.
Since Ω is convex there exists a real hyperplane H R such that H ⊂ H R and H R ∩ Ω = ∅ . By replacing P with e iθ P for some θ ∈ R we can assume that P ( H R ) = R and P (Ω) ⊂ H := { z ∈ C : Im( z ) > } . Then K Ω ( z , z ) ≥ K P (Ω) ( P ( z ) , P ( z )) ≥ K H ( P ( z ) , P ( z ))= 12 arcosh | P ( z ) − P ( z ) | P ( z )) Im( P ( z )) ! ≥
12 arcosh (cid:18) | P ( z ) | − | P ( z ) | ) | P ( z ) | | P ( z ) | (cid:19) = 12 arcosh (cid:18) | P ( z ) || P ( z ) | + | P ( z ) || P ( z ) | (cid:19) = 12 (cid:12)(cid:12)(cid:12)(cid:12) log (cid:12)(cid:12)(cid:12)(cid:12) P ( z ) P ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (cid:3) Since every point in the boundary of a convex domain is contained in a supportinghyperplane we have the following consequence of Lemma 2.7.
Lemma 2.8.
Suppose Ω ⊂ C d is a convex domain and x, y ∈ Ω are distinct. If L is the complex affine line containing x, y , then sup ξ ∈ L \ L ∩ Ω (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) k x − ξ kk y − ξ k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ K Ω ( x, y ) . Geodesics in complex hyperbolic space.
Let B d ⊂ C d be the unit ball.Then it is well known that ( B d , K B d ) is a standard model of complex hyperbolic d -space. In this subsection we describe some basic properties of geodesics in thismetric space, but first a definition. Definition 2.9. A complex geodesic in a domain Ω is a holomorphic map ϕ : D → Ωwhich satisfies K Ω ( ϕ ( z ) , ϕ ( w )) = K D ( z, w )for all z, w ∈ D .For the unit ball, every real geodesic is contained in a unique complex geodesic. BSTRUCTIONS TO BIHOLOMORPHISMS 9
Proposition 2.10. If γ : R ≥ → B d is a geodesic ray, then there exists a complexgeodesic ϕ : D → B d such that γ ( R ≥ ) ⊂ ϕ ( D ) . Moreover, ϕ is unique up toparametrization, that is: if ϕ : D → B d is a complex geodesic with γ ( R ≥ ) ⊂ ϕ ( D ) then ϕ = ϕ ◦ φ for some φ ∈ Aut( D ) . In the proof of Proposition 2.1 we will use the following fact about the asymptoticbehavior of geodesics in complex hyperbolic space.
Theorem 2.11. If γ , γ : R ≥ → B d are geodesic rays such that lim inf s,t →∞ K B d ( γ ( s ) , γ ( t )) < + ∞ , then there exists T ∈ R such that lim t →∞ K B d ( γ ( t ) , γ ( t + T )) = 0 . Moreover, if the images of γ and γ are contained in the same complex geodesic,then lim t →∞ t log K B d ( γ ( t ) , γ ( t + T )) = − otherwise lim t →∞ t log K B d ( γ ( t ) , γ ( t + T )) = − . Although this result is well known, we will sketch the proof of Theorem 2.11 inthe appendix.2.4.
The proof of Proposition 2.1.
Before starting the proof we state the fol-lowing observation:
Observation 2.12.
Suppose
C ⊂ C d is an open convex domain. If x + R ≥ · v ⊂ C for some x ∈ C and v ∈ C d , then x + R ≥ · v ⊂ C for every x ∈ C .Now for the rest of the subsection, suppose d ≥ C ⊂ C d is a convex domainwith the following properties:(1) C ∩
Span C { e , . . . , e d } = ∅ ,(2) C ∩ C · e = { ze : Im( z ) > } , and(3) C is biholomorphic to the unit ball.By Observation 2.12 and property (2) above, for every v ∈ Span C { e , . . . , e d } there exists some α v ∈ R ∪{∞} such that(1) { ze + v : Im( z ) > α v } = C ∩ (cid:16) C · e + v (cid:17) . Since
C ∩
Span C { e , . . . , e d } = ∅ we have that α v ∈ R ≥ ∪{∞} .Let S be the set of unit vectors in Span C { e , . . . , e d } . Then fix some δ > ie + 2 δ D · v ⊂ C for every v ∈ S . Let γ : R ≥ → C d be the curve given by γ ( t ) = e t ie and for v ∈ S let γ v : R ≥ → C d be the curve given by γ v ( t ) = δv + ( α δv + e t ) ie
10 OBSTRUCTIONS TO BIHOLOMORPHISMS
By Lemma 2.6 these curves are geodesic rays in ( C , K C ). Claim:
For every v ∈ S , lim t →∞ K C ( γ ( t ) , γ v ( t )) = 0 . Proof of Claim:
For t large let s t,v = t + 12 log (cid:16) − α δv e t (cid:17) . Then γ v ( s t,v ) = δv + e t ie and K C ( γ v ( t ) , γ v ( s t,v )) = 12 (cid:12)(cid:12)(cid:12) log (cid:16) − α δv e t (cid:17)(cid:12)(cid:12)(cid:12) . Since ie + 2 δ D · v ⊂ C , the equality in (1) implies that ire + 2 δ D · v ⊂ C for all r ≥
1. Hencelim sup t →∞ K C ( γ ( t ) , γ v ( t )) ≤ lim sup t →∞ (cid:16) K C ( γ ( t ) , γ v ( s t,v )) + K C ( γ v ( s t,v ) , γ v ( t )) (cid:17) = lim sup t →∞ K C ( γ ( t ) , γ v ( s t,v )) ≤ lim sup t →∞ K δ D (0 , δ ) < ∞ . Thus by Theorem 2.11 there exists T v ∈ R such thatlim t →∞ K C ( γ ( t ) , γ v ( t + T v )) = 0 . We claim that T v = 0. Let P : C d → C be the complex linear map given by P ( z , . . . , z d ) = z . Then by Lemma 2.70 = lim t →∞ K C ( γ ( t ) , γ v ( t + T v )) ≥ lim t →∞ (cid:12)(cid:12)(cid:12)(cid:12) log (cid:12)(cid:12)(cid:12)(cid:12) P ( γ ( t )) P ( γ v ( t + T v )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = lim t →∞ (cid:12)(cid:12)(cid:12)(cid:12) log e t e t + T v ) + α δv (cid:12)(cid:12)(cid:12)(cid:12) = | T v | and so T v = 0. (cid:3) By Lemma 2.6, for each v ∈ S the geodesics γ and γ v are contained in differentcomplex geodesics. So by Theorem 2.11 for each v ∈ S we havelim t →∞ t log K C ( γ ( t ) , γ v ( t )) = − . Moreover | K C ( γ ( t ) , γ v ( t )) − K C ( γ ( t ) , γ v ( s t,v )) | ≤ K C ( γ v ( t ) , γ v ( s t,v )) = 12 (cid:12)(cid:12)(cid:12) log (cid:16) − α δv e t (cid:17)(cid:12)(cid:12)(cid:12) = α δv e − t + O (cid:0) e − t (cid:1) . So we also have lim t →∞ t log K C ( γ ( t ) , γ v ( s t,v )) = − . Claim:
For every v ∈ S , lim sup t →∞ t log δ C ( e t ie ; v ) ≤ / BSTRUCTIONS TO BIHOLOMORPHISMS 11
Proof of Claim:
Note that K C ( γ ( t ) , γ v ( s t,v )) ≤ K δ C ( e t ie ; v ) D (0 , δ ) = K D (cid:18) , δ C ( e t ie ; v ) δ (cid:19) . Then since K D is locally Lipschitz on D × D and δ C ( e t ie ; v ) ≥ δ , there existssome C ≥ K C ( γ ( t ) , γ v ( s t,v )) ≤ C δδ C ( e t ie ; v ) . Hence − t →∞ t log K C ( γ ( t ) , γ v ( s t,v )) ≤ lim inf t →∞ − t log δ C ( e t ie ; v ) = − lim sup t →∞ t log δ C ( e t ie ; v )= − t →∞ t log δ C ( e t ie ; v ) (cid:3) Claim:
For every v ∈ S , lim inf t →∞ t log δ C ( e t ie ; v ) ≥ / . Proof of Claim:
Fix a sequence t n → ∞ such thatlim inf t →∞ t log δ C ( e t ie ; v ) = lim n →∞ t n log δ C ( e t n ie ; v ) . Then let z n ∈ C be such that | z n | = δ C ( e t n ie ; v ) and e t n ie + z n v ∈ ∂ C . Bypassing to a subsequence we can suppose that z n | z n | → e iθ for some θ ∈ R .Let v = − e iθ v , then by Lemma 2.8 we have K C ( γ ( t n ) , γ v ( s t n ,v )) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log (cid:13)(cid:13) γ ( t n ) − ( e t n ie + z n v ) (cid:13)(cid:13) k γ v ( s t n ,v ) − ( e t n ie + z n v ) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 (cid:12)(cid:12)(cid:12)(cid:12) log | z n || δe iθ + z n | (cid:12)(cid:12)(cid:12)(cid:12) . Now for n large (cid:12)(cid:12) δe iθ + z n (cid:12)(cid:12) > | z n | + δ/ n large K C ( γ ( t n ) , γ v ( s t n ,v )) ≥ (cid:12)(cid:12)(cid:12)(cid:12) log | z n || δe iθ + z n | (cid:12)(cid:12)(cid:12)(cid:12) = 12 log (cid:12)(cid:12) δe iθ + z n (cid:12)(cid:12) | z n | ≥
12 log (cid:18) δ | z n | (cid:19) . Since lim n →∞ K C ( γ ( t n ) , γ v ( s t n ,v )) = 0 , the above estimate implies that | z n | → ∞ , then using the fact that log : R > → R is locally bi-Lipschitz there exists some C > K C ( γ ( t n ) , γ v ( s t n ,v )) ≥ C | z n | = Cδ C ( e t n ie ; v ) . Hence − t →∞ t log K C ( γ ( t ) , γ v ( s t,v )) ≥ lim sup n →∞ t n log Cδ C ( e t n ie ; v ) = − lim inf n →∞ t n log δ C ( e t n ie ; v )= − t →∞ t log δ C ( e t ie ; v ) . (cid:3) The space of convex domains and the action of the affine group
Following work of Frankel [Fra89, Fra91], in this section we describe some factsabout the space of convex domains and the action of the affine group on this space.
Definition 3.1.
Let X d be the set of convex domains in C d which do not contain acomplex affine line and let X d, be the set of pairs (Ω , x ) where Ω ∈ X d and x ∈ Ω. Remark . The motivation for only considering convex domains which do notcontain complex affine lines comes from Theorem 2.4.We now describe a natural topology on the sets X d and X d, . Given two compactsets A, B ⊂ C d define the Hausdorff distance between them to be d H ( A, B ) = max (cid:26) max a ∈ A min b ∈ B k a − b k , max b ∈ B min a ∈ A k b − a k (cid:27) . The Hausdorff distance is a complete metric on the set of compact subsets of C d . Toconsider general closed sets, we introduce the local Hausdorff semi-norms betweentwo closed sets A, B ⊂ C d by defining d ( R ) H ( A, B ) = d H (cid:16) A ∩ B d (0; R ) , B ∩ B d (0; R ) (cid:17) for R >
0. Since an open convex set is determined by its closure, we can define atopology on X d and X d, using these seminorms:(1) A sequence C n ∈ X d converges to C ∈ X d if there exists some R ≥ d ( R ) H ( C n , C ) → R ≥ R ,(2) A sequence ( C n , x n ) ∈ X d, converges to ( C , x ) ∈ X d, if C n converges to C in X d and x n converges to x in C d .Let Aff( C d ) be the group of complex affine isomorphisms of C d . Then Aff( C d )acts on X d and X d, . Remarkably, the action of Aff( C d ) on X d, is co-compact: Theorem 3.3 (Frankel [Fra91]) . The group
Aff( C d ) acts co-compactly on X d, ,that is there exists a compact set K ⊂ X d, such that Aff( C d ) · K = X d, . Given some C and a sequence of points x n ∈ C the above theorem says that wecan find affine maps A n ∈ Aff( C d ) such that { A n ( C , x n ) } n ∈ N is relatively compactin X d, . Hence there exists a subsequence n k → ∞ such that A n k ( C , x n k ) convergesin X d, . Many of the arguments that follow rely on analyzing the geometry of thedomains obtained by this “rescaling” which leads to the next definition. Definition 3.4.
Given some
C ∈ X d let BlowUp( C ) ⊂ X d denote the set of C ∞ in X d where there exist a sequence x n ∈ C , a point x ∞ ∈ C ∞ , and affine maps A n ∈ Aff( C d ) such that BSTRUCTIONS TO BIHOLOMORPHISMS 13 (1) x n → ∞ in C (that is, for every compact subset K ⊂ C there exists some N > x n / ∈ K for all n ≥ N ),(2) A n ( C , x n ) converges to ( C ∞ , x ∞ ).For some domains, the set BlowUp( C ) is very special. Proposition 3.5.
Suppose that
C ⊂ C d is a convex domain which is biholomorphicto a bounded strongly pseudoconvex domain with C boundary. Then every C ∞ ∈ BlowUp( C ) is biholomorphic to the unit ball in C d . This is a consequence of the Frankel-Pinchuk rescaling method, but we willprovide a proof using the squeezing function.
Proof.
Suppose that C ∞ ∈ BlowUp( C ). Then fix a sequence x n ∈ C such that x n → ∞ in C , a point x ∞ ∈ C ∞ , and affine maps A n ∈ Aff( C d ) such that A n ( C , x n )converges to ( C ∞ , x ∞ ).By results of Diederich, Fornæss, and Wold [DFW14, Theorem 1.1] and Deng,Guan, and Zhang [DGZ16, Theorem 1.1] (see Theorem 1.6 above)lim n →∞ s C ( x n ) = 1 . Now the function (Ω , x ) ∈ X d, → s Ω ( x ) is an upper semicontinuous function on X d, (see Proposition 7.1 in [Zim16]). So s C ∞ ( x ∞ ) ≥ lim n →∞ s A n C ( A n x n ) = lim n →∞ s C ( x n ) = 1 . Then s C ∞ ( x ∞ ) = 1 and so C ∞ is biholomorphic to the unit ball in C d by Theorem2.1 in [DGZ12]. (cid:3) We next define a particular compact subset of X d, whose Aff( C d )-translatescover X d, . Recall that D = { z ∈ C : | Im( z ) | + | Re( z ) | < } . For 1 ≤ i ≤ d consider the complex ( d − i )-dimensional affine plane Z i = e i + Span C { e i +1 , . . . , e d } . Definition 3.6.
Let K d ⊂ X d be the set of convex domains Ω such that:(1) D e i ⊂ Ω for each 1 ≤ i ≤ d ,(2) Z i ∩ Ω = ∅ for each 1 ≤ i ≤ d .Also let K d, = { (Ω ,
0) : Ω ∈ K } . Theorem 3.7. [Zim16, Theorem 2.5]
With the notation above: K d, is a compactsubset of X d, and Aff( C d ) · K d, = X d, .Remark . In [Zim16] the set K d ⊂ X d was slightly different: in particular onehad the requirement that D e i ⊂ Ω for each 1 ≤ i ≤ d instead of D e i ⊂ Ω for each 1 ≤ i ≤ d. However, the proof is identical.We end this section with a technical result which will allow us to reduce calcu-lations to the two dimensional case.
Proposition 3.9.
Suppose
C ∈ X d is a convex domain such that: (1) C ∩ ( e + Span C { e , . . . , e d } ) = ∅ and (2) C ∩
Span C { e , e } ∈ K ,then there exists A ∈ GL d ( C ) such that A | Span C { e ,e } = Id Span C { e ,e } and A C ∈ K d .Proof. We will select points ξ , . . . , ξ d ∈ ∂ C and subspaces H , . . . , H d ⊂ C d asfollows. First let ξ = e and H = Span C { e , . . . , e d } . Then let ξ = e and let H be a ( d − e + H ) ∩ C = ∅ and H ⊂ H = Span C { e , . . . , e d } . Since Span C { e , . . . , e d } ∩ C is convex and e ∈ ∂ C , such a subspace exists. Nowsupposing that ξ , . . . , ξ k − and H , . . . , H k − have already been selected, we pick ξ k and H k as follows: let ξ k be a point in H k − ∩ ∂ C closest to 0 and let H k be a( d − k )-dimensional complex subspace such that H k ⊂ H k − and ( ξ k + H k ) ∩ C = ∅ .Since H k − ∩ C is convex and ξ k ∈ ∂ ( H k − ∩ C ), such a subspace exists.Notice that(1) C · ξ k + H k = H k − for k ≥ H k = Span C { ξ k +1 , . . . , ξ d } for k ≥
1, and(3) Span C { ξ , . . . , ξ d } = C d .Now let A ∈ GL d ( C ) be the complex linear map with A ( ξ i ) = e i for 1 ≤ i ≤ d .Since ξ , . . . , ξ d is a basis of C d , the linear map A is well defined. Since ξ = e and ξ = e we see that A | Span C { e ,e } = Id Span C { e ,e } .We now claim that A C ∈ K d . Since A C ∩
Span C { e , e } ∈ K we have D · e i ⊂ A C for i = 1 , D · e i ⊂ A C for i = 3 , . . . , d. So D · e i ⊂ A C for i = 1 , . . . , d. Since A ( ξ k ) = e k and H k = Span C { ξ k +1 , . . . , ξ d } we have A C ∩ Z k = A (cid:0) C ∩ A − Z k (cid:1) = A ( C ∩ ( ξ k + Span C { ξ k +1 , . . . , ξ d } ))= A ( C ∩ ( ξ k + H k )) = ∅ . So A C ∈ K d . (cid:3) The proof of Theorem 1.4
In this section we will prove Theorem 1.4 which we begin by recalling.
Theorem 4.1.
Suppose Ω ⊂ C d is a bounded strongly pseudoconvex domain with C boundary and C ⊂ C d is a convex domain biholomorphic to Ω . If C has C boundary, then for every ǫ > and R > there exists a C = C ( ǫ, R ) ≥ such that δ C ( z ; v ) ≤ Cδ C ( z ) / (2+ ǫ ) for all z ∈ C with k z k ≤ R and all nonzero v ∈ C d . BSTRUCTIONS TO BIHOLOMORPHISMS 15
For the rest of the section, fix a convex domain
C ⊂ C d satisfying the conditionsof the theorem. Then fix some ǫ > R > z ∈ C let P z be the set of points in ∂ C which are closest to z . Then pick R ′ ≥ R such that P z ⊂ B d (0; R ′ )for all z ∈ B d (0; R ) ∩ C . Next let K = B d (0; R ′ ) ∩ ∂ C . For ξ ∈ ∂ C let n ( ξ ) be theinward pointing unit normal vector of C at ξ . Finally fix δ ∈ (0 ,
1) such that ξ + r n ( ξ ) ∈ C for all ξ ∈ K and r ∈ (0 , δ ]. As before let D = { z ∈ C : | Im( z ) | + | Re( z ) | < } . Since ∂ C is C , by shrinking δ > ξ + δ n ( ξ ) + δ D · n ( ξ ) ⊂ C for all ξ ∈ K . Then ξ + r n ( ξ ) + r D · n ( ξ ) ⊂ ξ + δ n ( ξ ) + δ D · n ( ξ ) ⊂ C for all ξ ∈ K and r ∈ (0 , δ ].We begin by showing that the desired estimate holds for tangential directions. Lemma 4.2.
With the notation above, there exists C > such that δ C ( ξ + r n ( ξ ); v ) ≤ C r / (2+ ǫ ) for all ξ ∈ K , r ∈ (0 , δ ] , and nonzero v ∈ T C ξ ∂ C .Proof. Suppose not, then there exist ξ n ∈ K , r n ∈ (0 , δ ], unit vectors v n ∈ T C ξ n ∂ C ,and C n > C n → ∞ and δ C ( ξ n + r n n ( ξ n ); v n ) = C n r / (2+ ǫ ) n . By increasing r n if necessary we can assume in addition that δ C ( ξ n + r n ( ξ n ); v n ) ≤ C n r / (2+ ǫ ) for all r ∈ [ r n , δ ]. Since C contains no complex affine lines, we must have r n → n , let τ n : C d → C d be an affine isometry such that(1) τ n ( ξ n ) = 0,(2) τ n ( ξ n + n ( ξ n )) = ie ,(3) τ n ( ξ n + v n ) = e .Conditions (1) and (2) imply that T τ n ( ∂ C ) = { ( z , . . . , z n ) ∈ C n : Im( z ) = 0 } and τ n ( C ) ⊂ { ( z , . . . , z n ) ∈ C n : Im( z ) > } . Condition (3) implies that δ τ n ( C ) ( r n ie ; e ) = C n r / (2+ ǫ ) n . Then pick z n ∈ C such that | z n | = C n r / (2+ ǫ ) n and r n ie + z n e ∈ ∂τ n C . Then consider the diagonal matrix A n = r n z n . Let C n = A n τ n ( C ). Since ξ n + r n n ( ξ n ) + r n D · n ( ξ n ) ⊂ C we see that ie + D · e ⊂ C n . Further, by construction:(1) { ( z , . . . , z n ) ∈ C n : Im( z ) = 0 } ∩ C n = ∅ ,(2) ie + e / ∈ C n , and(3) ie + D · e ⊂ C n .Hence C n ∩ Span C { e , e } ⊂ ie + K where K ⊂ X is the subset from Defini-tion 3.6. Now by Proposition 3.9 there exists an affine map B n ∈ Aff( C d ) such that B n | Span C { e ,e } = Id Span C { e ,e } and B n C n ∈ ie + K d .Now since K d is compact in X d , we can pass to a subsequence such that B n C n converges to some C ∞ in X d . Notice that B n C n = B n A n τ n C and ie = ( B n A n τ n )( ξ n + r n n ( ξ n )) . Since r n → ie ∈ C ∞ we see that C ∞ ∈ BlowUp( C ) . We next claim that C ∞ satisfies conditions (1), (2), and (3) from Proposition 2.1.By Proposition 3.5, C ∞ is biholomorphic to the unit ball and hence satisfies condi-tion (3).Since each B n C n is in ie + K d , we see that { ( z , . . . , z n ) ∈ C n : Im( z ) = 0 } ∩ B n C n = ∅ and so { ( z , . . . , z n ) ∈ C n : Im( z ) = 0 } ∩ C ∞ = ∅ . Hence C ∞ satisfies condition (1).For η > r ∈ (0 , ∞ ] let A ( r ; η ) = { z ∈ C : 0 < | z | < r and | Im( z ) | < η Re( z ) } . Since
K ⊂ ∂ C is compact and ∂ C is a C hypersurface, for any η > r η > ξ + A ( r η ; η ) · n ( ξ ) ⊂ C for all ξ ∈ K . Then for any η > A ( r η /r n ; η ) · ie ⊂ B n C n and so A ( ∞ ; η ) · ie ⊂ C ∞ . BSTRUCTIONS TO BIHOLOMORPHISMS 17
Since η > C ∞ ⊂ { ( z , . . . , z d ) ∈ C d : Im( z ) > } we then have { ze : Im( z ) > } = C ∞ ∩ C · e . Hence C ∞ satisfies condition (2).However, if 1 ≤ r ≤ δ/r n , then δ B n C n ( rie ; e ) = δ C n ( rie ; e ) = 1 | z n | δ τ n ( C ) ( r n rie ; e )= 1 | z n | δ C ( ξ + r n r n ( ξ n ); v n ) ≤ | z n | C n ( r n r ) / (2+ ǫ ) = r / (2+ ǫ ) . So for 1 ≤ r we have δ C ∞ ( rie ; e ) ≤ r / (2+ ǫ ) . Which Proposition 2.1 says is impossible. So we have a contradiction. (cid:3)
We now prove the desired estimate for all directions.
Lemma 4.3.
With the notation above, there exists C ≥ such that δ C ( x ; v ) ≤ Cδ C ( x ) / (2+ ǫ ) for all x ∈ B d (0; R ) ∩ C and all nonzero v ∈ C d .Proof. Since C does not contain any complex affine lines, there exists M > δ C ( x ; v ) ≤ M for all x ∈ B d (0; R ) ∩ C and all nonzero v ∈ C d . Next let K d ⊂ X d be the subsetfrom Definition 3.6. Since K d ⊂ X d is compact there exists C > δ C ′ (0; v ) ≤ C for all C ′ ∈ K d and nonzero v ∈ C d .We claim that δ C ( x ; v ) ≤ max n M δ − / (2+ ǫ ) , C C o δ C ( x ) / (2+ ǫ ) for all x ∈ B d (0; R ) ∩ C and all nonzero v ∈ C d .Fix x ∈ C . If δ C ( x ) ≥ δ then δ C ( x ; v ) ≤ M ≤ M (cid:18) δ C ( x ) δ (cid:19) / (2+ ǫ ) ≤ Cδ C ( x ) / (2+ ǫ ) for all nonzero v ∈ C d . So suppose that δ C ( x ) < δ . Let ξ ∈ ∂ C be a point in ∂ C closest to x . Then x = ξ + δ C ( x ) n ( ξ )and by construction ξ ∈ K .Next we pick points ξ , ξ , . . . , ξ d as follows. First let ξ = ξ . Next, assuming ξ , . . . , ξ k have been already selected let P k +1 be the ( d − k )-dimensional complexplane through x which is orthogonal to the lines xξ i . Then let ξ k +1 be a point in P k +1 ∩ ∂ C which is closest to x . By construction − x + P = T C ξ ∂ C and hence( ξ − x ) , . . . , ( ξ d − x ) ∈ T C ξ ∂ C . So by the lemma above δ C ( x ; ξ i − x ) ≤ C δ C ( x ) / (2+ ǫ ) for i ≥
2. Moreover, since C ≥ δ C ( x ) < δ ≤ δ C ( x ; ξ − x ) = δ C ( x ) ≤ C δ C ( x ) / (2+ ǫ ) . Next let τ : C d → C d be the affine translation τ ( z ) = z − x and let U be theunitary transformation such that U τ ( ξ i ) = δ C ( x ; ξ i − x ) e i . Then let Λ = δ C ( x ; ξ − x ) − . . . δ C ( x ; ξ d − x ) − . Finally let A be the affine map A = Λ U τ . Then we have A C ∈ K d . So if v ∈ C d isa unit vector, then δ C ( x ; v ) = 1 k Λ U v k δ A C (0; Λ U v ) ≤ C k Λ U v k ≤ C C δ C ( x ) / (2+ ǫ ) since k Λ U v k ≥ k Λ − k k v k ≥ C δ C ( x ) / (2+ ǫ ) . (cid:3) Characterizing strong pseudoconvexity
Theorems 1.7, 1.10, and 1.11 are particular cases of more general theorems whichwe now describe. In order to state these results we need to define intrinsic functionson the space of convex domains.
Definition 5.1.
A function f : X d, → R is called intrinsic if f ( C , x ) = f ( C , x )whenever there exists a biholomorphism ϕ : C → C with ϕ ( x ) = x . Example 5.2.
The functions: ( C , x ) → s C ( x )and ( C , x ) → max v ∈ T x C \{ } (cid:12)(cid:12)(cid:12)(cid:12) H b C ( v ) − − d + 1 (cid:12)(cid:12)(cid:12)(cid:12) are both intrinsic.Since the unit ball is a homogeneous domain we have the following: Observation 5.3. If B d ⊂ C d is the unit ball and f : X d, → R is an intrinsicfunction, then f ( B d , x ) = f ( B d ,
0) for all x ∈ B d .Recall that the set X d, has a topology coming from the local Hausdorff topology(see Section 3 above) and when an intrinsic function is continuous in this topology aversion of Klembeck’s Theorem (see Theorem 1.9 above) holds for convex domains: BSTRUCTIONS TO BIHOLOMORPHISMS 19
Proposition 5.4. [Zim16, Proposition 1.13]
Suppose f : X d, → R is a continuousintrinsic function and C is a bounded convex domain with C boundary. If ξ ∈ ∂ C is a strongly pseudoconvex point of ∂ C , then lim z → ξ f ( C , z ) = f ( B d , . We will prove the following two converses to the above proposition:
Theorem 5.5. (see Subsection 5.3) Suppose that f : X d, → R is a continuousintrinsic function with the following property: if C ∈ X d and f ( C , x ) = f ( B d , forall x ∈ C , then C is biholomorphic to B d .Then for any α > there exists some ǫ = ǫ ( d, f, α ) > such that: if C ⊂ C d isa bounded convex domain with C ,α boundary and | f ( C , z ) − f ( B d , | ≤ ǫ outside some compact subset of C , then C is strongly pseudoconvex and thus lim z → ∂ C f ( C , z ) = f ( B d , . Some interesting intrinsic functions, for instance the squeezing function, do notappear to be continuous on X d, but are upper-semicontinuous. So we will alsoestablish the following: Theorem 5.6. (see Subsection 5.4) Suppose that f : X d, → R is an upper semi-continuous intrinsic function with the following property: if C ∈ X d and f ( C , x ) ≥ f ( B d , for all x ∈ C , then C is biholomorphic to B d .Then for any α > there exists some ǫ = ǫ ( d, f, α ) > such that: if C ⊂ C d isa bounded convex domain with C ,α boundary and f ( C , z ) ≥ f ( B d , − ǫ outside some compact subset of C , then C is strongly pseudoconvex. An example of Fornæss and Wold.
In this subsection we will use anexample of Fornæss and Wold to show that Theorem 5.5 and Theorem 5.6 both failfor convex domains with C boundary. Proposition 5.7.
For any d ≥ there exists a bounded convex domain C ⊂ C d with C boundary which is not strongly pseudoconvex, but has the following properties: (1) If f : X d, → R is a continuous intrinsic function, then lim z → ∂ C f ( C , z ) = f ( B d , , (2) If f : X d, → R is an upper semi-continuous intrinsic function, then lim z → ∂ C f ( C , z ) ≥ f ( B d , , Proof.
For any d ≥
2, Fornæss and Wold [FW16] have constructed an exampleof a bounded convex domain
C ⊂ C d with C boundary which is not stronglypseudoconvex, but still satisfies lim z → ∂ C s C ( z ) = 1 . Now suppose that f : X d, → R is a continuous intrinsic function. We claim thatlim z → ∂ C f ( C , z ) = f ( B d , , Suppose not then there exist a boundary point ξ ∈ ∂ C and a sequence z n ∈ C suchthat z n → ξ and lim n →∞ f ( C , z n ) = f ( B d , . Now by Theorem 3.3 we can find affine maps A n ∈ Aff( C d ) such that A n ( C , z n )converges to some ( C ∞ , z ∞ ) ∈ X d, . Since the squeezing function is an upper semi-continuous function on X d, (see [Zim16, Proposition 7.1]) we have s C ∞ ( z ∞ ) ≥ lim sup n →∞ s A n C ( A n z n ) = lim sup n →∞ s C ( z n ) = 1 . So s C ∞ ( z ∞ ) = 1. Then C ∞ is biholomorphic to the unit ball by Theorem 2.1in [DGZ12]. Then since f is continuous and intrinsiclim n →∞ f ( C , z n ) = lim n →∞ f ( A n C , A n z n ) = f ( C ∞ , z ∞ ) = f ( B d , . So we have a contradiction.The proof of part (2) is essentially identical. (cid:3)
Rescaling revisited.
In this subsection we prove the following rescaling re-sult:
Proposition 5.8.
Suppose
C ⊂ C d is a convex domain which does not containany complex lines. If C has C ,α boundary for some α > and is not stronglypseudoconvex, then there exists some C ∞ ∈ BlowUp( C ) such that: (1) C ∞ ∈ ie + K d , (2) C ∞ ∩ Span C { e , . . . , e d } = ∅ , (3) C ∞ ∩ C · e = { ze : Im( z ) > } , and (4) δ C ∞ ( rie ; e ) ≤ r / (2+ α ) for r ≥ . The proof of the Proposition is very similar to the proof of Theorem 1.4, but wewill provide the details anyways.
Proof.
Since C is not strongly pseudoconvex, there exists a non-strongly pseudo-convex point ξ ∈ ∂ C . Then there exist C, δ > v ∈ T C ξ ∂ C suchthat δ C ( ξ + r n ( ξ ); v ) ≥ Cr / (2+ α ) for every r ∈ (0 , δ ]. Since ∂ C is C , by shrinking δ > ξ + r n ( ξ ) + r D · n ( ξ ) ⊂ C for r ∈ (0 , δ ].Then lim r → r / (2+ α )+ ǫ δ C ( ξ + r n ( ξ ); v ) = 0for every ǫ > ǫ n → r n → n →∞ r / (2+ α )+ ǫ n n δ C ( ξ + r n n ( ξ ); v ) = 0 . BSTRUCTIONS TO BIHOLOMORPHISMS 21
Then let C n > δ C ( ξ + r n n ( ξ ); v ) = C n r / (2+ α )+ ǫ n n . Since C n → ∞ , by increasing r n if necessary we can assume in addition that δ C ( ξ + r n ( ξ ); v ) ≤ C n r / (2+ α )+ ǫ n for all r ∈ [ r n , δ ]. Since C contains no complex affine lines, even after possiblyincreasing each r n we still have r n → τ ∈ Aff( C d ) be an affine isometry of C d such that(1) τ ( ξ ) = 0,(2) τ ( ξ + n ( ξ )) = ie , and(3) τ ( ξ + v ) = e .Notice that conditions (1) and (2) imply that T τ (Ω) = { ( z , . . . , z d ) ∈ C d : Im( z ) = 0 } and τ (Ω) ⊂ { ( z , . . . , z d ) ∈ C d : Im( z ) > } . Condition (3) implies that δ τ (Ω) ( r n ie ; e ) = C n r / (2+ α )+ ǫ n n . Then pick z n ∈ C such that | z n | = C n r / (2+ α )+ ǫ n n and r n ie + z n e ∈ ∂T C . Then consider the diagonal matrix A n = r n z n . Let C n = A n τ ( C ). Since ξ + r n n ( ξ ) + r n D · n ( ξ ) ⊂ C , we have ie + D · e ⊂ C n . Further, by construction:(1) { ( z , . . . , z n ) ∈ C n : Im( z ) = 0 } ∩ C n = ∅ ,(2) ie + e / ∈ C n , and(3) ie + D · e ⊂ C n .Hence C n ∩ Span C { e , e } ∈ ie + K where K ⊂ X is the subset from Defini-tion 3.6. By Proposition 3.9 there exists an affine map B n ∈ Aff( C d ) such that B n | Span C { e ,e } = Id Span C { e ,e } and B n C n ∈ ie + K d .Now since K d is compact in X d , we can pass to a subsequence such that B n C n converges to some C ∞ in X d . Notice that B n C n = B n A n τ C and ie = ( B n A n τ )( ξ + r n n ( ξ )) . Since ξ + r n n ( ξ ) converges to the boundary of C and ie ∈ C ∞ we see that C ∞ ∈ BlowUp( C ) . Moreover, by construction C ∞ ∈ K d and C ∞ ∩ Span C { e , . . . , e d } = ∅ .As in the proof of Theorem 1.4 for η > r ∈ (0 , ∞ ] let A ( r ; η ) = { z ∈ C : 0 < | z | < r and | Im( z ) | < η Re( z ) } . Since ∂ C is a C hypersurface, for any η > r η > ξ + A ( r η ; η ) · n ( ξ ) ⊂ C . Then for any η > A ( r η /r n ; η ) · ie ⊂ B n A n τ ( C )and so A ( ∞ ; η ) · ie ⊂ C ∞ . Since η > C ∞ ⊂ { ( z , . . . , z d ) ∈ C d : Im( z ) > } we then have C ∞ ∩ C · e = { ze : Im( z ) > } .Finally, if 1 ≤ r ≤ δ/r n , then δ B n C n ( rie ; e ) = δ C n ( rie ; e ) = 1 | z n | δ τ ( C ) ( r n rie ; e )= 1 | z n | δ C ( ξ + r n r n ( ξ ); v ) ≤ | z n | C n ( r n r ) / (2+ α )+ ǫ n = r / (2+ α )+ ǫ n . So for 1 ≤ r we have δ C ∞ ( rie ; e ) ≤ r / (2+ α ) . (cid:3) The proof of Theorem 5.5.
Fix d ≥
2, a continuous intrinsic function f : X d, → R satisfying the hypothesis of the theorem, and some α >
0. Supposefor a contradiction that there exists a sequence of convex domains C n ∈ X d, suchthat:(1) each C n has C ,α boundary,(2) each C n is not strongly pseudoconvex, and(3) for all n ∈ N | f ( C n , z ) − f ( B d , | ≤ /n outside some compact subset of C n .Now using Proposition 5.8 for each n we can find some C n, ∞ ∈ BlowUp( C n ) suchthat(1) C n, ∞ ∈ ie + K d ,(2) C n, ∞ ∩ Span C { e , . . . , e d } = ∅ ,(3) C n, ∞ ∩ C · e = { ze : Im( z ) > } , and(4) δ C n, ∞ ( e r ie ; e ) ≤ e r/ (2+ α ) for r ≥ BSTRUCTIONS TO BIHOLOMORPHISMS 23
We claim that | f ( C n, ∞ , z ) − f ( B d , | ≤ /n for all z ∈ C n, ∞ . By the definition of BlowUp( C n ), there exist a sequence x m ∈ C n ,a point x ∞ ∈ C n, ∞ , and affine maps A m ∈ Aff( C d ) such that x m → ∞ in C n and A m ( C n , x m ) converges to ( C n, ∞ , x ∞ ). Now fix z ∈ C n, ∞ and a relatively compactconvex subdomain O ⊂ C n, ∞ which contains x ∞ and z . By the definition of thelocal Hausdorff topology, O ⊂ A m C n for m sufficiently large. So for m sufficientlylarge A − m ( O ) ⊂ C n . Then K C n ( x m , A − m z ) ≤ K A − m O ( x m , A − m z ) = K O ( A m x m , z )and since A m x m → x ∞ we see thatlim sup m →∞ K C n ( x m , A − m z ) < ∞ . Since K C n is a proper metric on C n and x m approaches the boundary of C n , wesee that A − m z approaches the boundary of C n . But then, since f is continuous andintrinsic, | f ( C n, ∞ , z ) − f ( B d , | = lim m →∞ (cid:12)(cid:12) f ( C n , A − m z ) − f ( B d , (cid:12)(cid:12) ≤ /n. Now since K d ⊂ X d is compact, we can pass to a subsequence such that C n, ∞ converges in X d to some convex domain C ∞ . Since f is continuous, we see that f ( C ∞ , z ) = f ( B d , z ∈ C ∞ . So by hypothesis C ∞ is biholomorphic to the unit ball. On theother hand, by the definition of the local Hausdorff topology, we see that(1) C ∞ ∩ Span C { e , . . . , e d } = ∅ ,(2) C ∞ ∩ C · e = { ze : Im( z ) > } , and(3) δ C ∞ ( e r ie ; e ) ≤ e r/ (2+ α ) for r ≥ The proof of Theorem 5.6.
This is essentially identical to the proof ofTheorem 5.5.5.5.
The proof of Theorem 1.7.
The function ( C , x ) ∈ X d, → s C ( x ) is an uppersemicontinuous intrinsic function (see [Zim16, Proposition 7.1]) and by Theorem2.1 in [DGZ12] if s Ω ( x ) = 1 for some x ∈ Ω, then Ω is biholomorphic to the unitball. Hence Theorem 1.7 follows from Theorem 5.6.5.6.
K¨ahler metrics with controlled geometry.
We begin by introducing thefollowing class of metrics on a domain which are informally the K¨ahler metricswhich have controlled geometry relative to the Kobayashi metic.
Definition 5.9.
Suppose Ω ⊂ C d is a bounded domain and M >
1. Let G M (Ω) bethe set of K¨aher metrics g on Ω (with respect to the standard complex structure)with the following properties:(1) g is a C metric,(2) For all z ∈ Ω and v ∈ C d ,1 M p g z ( v, v ) ≤ k Ω ( z ; v ) ≤ M p g z ( v, v ) . (3) If X, v, w ∈ C d , then | X ( g z ( v, w )) | ≤ M k Ω ( z ; X ) k Ω ( z ; v ) k Ω ( z ; w ) . (4) If X, Y, v, w ∈ C d , then | Y ( X ( g z ( v, w ))) | ≤ M k Ω ( z ; Y ) k Ω ( z ; X ) k Ω ( z ; v ) k Ω ( z ; w ) . (5) If X, Y, v, w ∈ C d and z , z ∈ Ω, then | Y ( X ( g z ( v, w ))) − Y ( X ( g z ( v, w ))) |≤ M k Ω ( z ; Y ) k Ω ( z ; X ) k Ω ( z ; v ) k Ω ( z ; w ) K Ω ( z , z ) . Definition 5.10.
For
M, d >
0, define a function h M : X d, → R by letting h M ( C , x ) be the infimum of all numbers ǫ > g ∈ G M ( C ) with max v,w ∈ T z C \{ } | H g ( v ) − H g ( w ) | ≤ ǫ for all z ∈ B C ( x ; 1 /ǫ )where B C ( x ; r ) is the closed ball of radius r about the point x ∈ C with respect tothe Kobayashi distance.In [Zim16, Proposition 8.2, 8.3] we proved that − h M is an upper semi-continuousintrinsic function on X d, and if h M ( C , x ) = 0 for some x ∈ C then C is biholomorphicto the unit ball in C d . So Theorem 5.6 implies the following: Corollary 5.11.
For any d, M, α > there exists ǫ = ǫ ( d, M, α ) > such that: if C ⊂ C d is a bounded convex domain with C ,α boundary and h M ( C , z ) ≤ ǫ outside some compact subset of C , then C is strongly pseudoconvex. Theorem 1.11 is now a simple consequence of this result.
Proof of Theorem 1.11.
Fix ǫ >
C ⊂ C d is abounded convex domain with C ,α boundary and h M ( C , z ) ≤ ǫ outside a compact set of C , then C is strongly pseudoconvex.Now suppose that C ⊂ C d is a bounded convex domain with C ,α boundary, K ⊂ C is compact, and there exists a metric g ∈ G M ( C ) such thatmax v,w ∈ T z C \{ } | H g ( v ) − H g ( w ) | ≤ ǫ for all z ∈ C \ K. We claim that C is strongly pseudoconvex.Since K C is a proper distance on C (see Theorem 2.4), there exists some compactsubset K ′ ⊂ C such that B C ( x ; 1 / (2 ǫ )) ⊂ C \ K for all x ∈ C \ K ′ . Then, with thischoice of K ′ , h M ( C , x ) ≤ ǫ for all x ∈ C \ K ′ . So by our choice of ǫ > C is strongly pseudoconvex. (cid:3) The proof of Theorem 1.10.
In [Zim16, Proposition 9.1] we proved that forany d > M = M ( d ) > C ∈ X d then b C ∈ G M ( C )for all M ≥ M . So Theorem 1.10 is a corollary of Theorem 1.11. BSTRUCTIONS TO BIHOLOMORPHISMS 25
Appendix A. Properties of complex hyperbolic space
In this section we sketch the proof of Theorem 2.11:
Theorem A.1. If γ , γ : R ≥ → B d are geodesic rays such that lim inf s,t →∞ K B d ( γ ( s ) , γ ( t )) < + ∞ , then there exists T ∈ R such that lim t →∞ K B d ( γ ( t ) , γ ( t + T )) = 0 . Moreover, if the images of γ and γ are contained in the same complex geodesicthen lim t →∞ t log K B d ( γ ( t ) , γ ( t + T )) = − otherwise lim t →∞ t log K B d ( γ ( t ) , γ ( t + T )) = − . Proof.
The first assertion is a consequence of the Kobayashi distance on B d be-ing induced by a negatively curved Riemannian metric (it is isometric to complexhyperbolic space), see for instance [HIH77, Proposition 4.1].To establish the second assertion it is easiest to work with the domain P d = ( ( z , . . . , z d ) : Im( z ) > d X i =2 | z i | ) which is biholomorphic to B d .Suppose that γ , γ : R ≥ → P d are geodesic rays withlim t →∞ K P d ( γ ( t ) , γ ( t )) = 0 . Using the fact that the biholomorphism group Aut ( P d ) of P d acts transitively onthe set of geodesic rays in P d , we can assume that γ ( t ) = ie t e . Then we must have γ ( t ) = v + (cid:16) α + i ( e t + k v k ) (cid:17) e for some v ∈ Span C { e , . . . , e d } and α ∈ R . Moreover, γ and γ are contained inthe same complex geodesic if and only if v = 0.The estimates on lim t →∞ t log K P d ( γ ( t ) , γ ( t ))will follow from the well known fact that if V ⊂ C d is an affine subspace whichintersects P d then K V ∩P d ( z, w ) = K P d ( z, w )for all z, w ∈ V ∩ P d .First suppose that v = 0. Thenlim t →∞ t log K P d ( γ ( t ) , γ ( t )) = lim t →∞ t log K H ( ie t , α + ie t ) where H = { z ∈ C : Im( z ) > } . Then K H ( ie t , α + ie t ) = 12 arcosh (cid:18) α e t (cid:19) and using the fact that arcosh( x ) = log( x + √ x −
1) we then have K H ( ie t , α + ie t ) = 12 log (cid:18) α e t + | α |√ e t (cid:19) = | α | √ e − t + O (cid:0) e − t (cid:1) . So lim t →∞ t log K P d ( γ ( t ) , γ ( t )) = − . Next suppose that v = 0. Then let γ ( t ) = v + i ( e t + k v k ) e . Sincelim t →∞ t log K P d ( γ ( t ) , γ ( t )) = − t →∞ t log K P d ( γ ( t ) , γ ( t )) = − . Next for t sufficiently large let s t = t + 12 log − k v k e t ! . Then K P d ( γ ( t ) , γ ( s t )) = 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log − k v k e t !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k v k e − t + O (cid:0) e − t (cid:1) so it is enough to show thatlim t →∞ t log K P d ( γ ( t ) , γ ( s t )) = − . Now since γ ( s t ) = v + ie t e and P d ∩ (cid:0) ie t + C · v (cid:1) = (cid:26) ie t + z v k v k : z ∈ C , | z | ≤ e t (cid:27) . we havelim t →∞ t log K P d ( γ ( t ) , γ ( s t )) = lim t →∞ t log K e t D (0 , k v k ) = lim t →∞ t log K D (0 , e − t k v k ) = − K D (0 , z ) = | z | + O (cid:16) | z | (cid:17) for z closeto 0. (cid:3) References [Bar80] Theodore J. Barth. Convex domains and Kobayashi hyperbolicity.
Proc. Amer. Math.Soc. , 79(4):556–558, 1980.[Bel81] Steven R. Bell. Biholomorphic mappings and the ¯ ∂ -problem. Ann. of Math. (2) ,114(1):103–113, 1981.[BH99] Martin R. Bridson and Andr´e Haefliger.
Metric spaces of non-positive curvature , vol-ume 319 of
Grundlehren der Mathematischen Wissenschaften [Fundamental Principlesof Mathematical Sciences] . Springer-Verlag, Berlin, 1999.
BSTRUCTIONS TO BIHOLOMORPHISMS 27 [Bla85] John S. Bland. On the existence of bounded holomorphic functions on complete K¨ahlermanifolds.
Invent. Math. , 81(3):555–566, 1985.[Bla89] John S. Bland. Bounded imbeddings of open K¨ahler manifolds in C N . Duke Math. J. ,58(1):173–203, 1989.[DFW14] K. Diederich, J. E. Fornæss, and E. F. Wold. Exposing points on the boundary of astrictly pseudoconvex or a locally convexifiable domain of finite 1-type.
J. Geom. Anal. ,24(4):2124–2134, 2014.[DGZ12] Fusheng Deng, Qian Guan, and Liyou Zhang. Some properties of squeezing functions onbounded domains.
Pacific J. Math. , 257(2):319–341, 2012.[DGZ16] Fusheng Deng, Qi’an Guan, and Liyou Zhang. Properties of squeezing functions andglobal transformations of bounded domains.
Trans. Amer. Math. Soc. , 368(4):2679–2696,2016.[Fra89] Sidney Frankel. Affine approach to complex geometry. In
Recent developments in geom-etry (Los Angeles, CA, 1987) , volume 101 of
Contemp. Math. , pages 263–286. Amer.Math. Soc., Providence, RI, 1989.[Fra91] Sidney Frankel. Applications of affine geometry to geometric function theory in severalcomplex variables. I. Convergent rescalings and intrinsic quasi-isometric structure. In
Sev-eral complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989) , volume 52of
Proc. Sympos. Pure Math. , pages 183–208. Amer. Math. Soc., Providence, RI, 1991.[FW16] J. E. Fornæss and E.F. Wold. A non-strictly pseudoconvex domain for which the squeezingfunction tends to one towards the boundary. To appear in
Pacific J. Math. , 2016.[Haw53] N. S. Hawley. Constant holomorphic curvature.
Canadian J. Math. , 5:53–56, 1953.[HIH77] Ernst Heintze and Hans-Christoph Im Hof. Geometry of horospheres.
J. DifferentialGeom. , 12(4):481–491 (1978), 1977.[Igu54] Jun-ichi Igusa. On the structure of a certain class of Kaehler varieties.
Amer. J. Math. ,76:669–678, 1954.[JK16] S. Joo and K.-T. Kim. On boundary points at which the squeezing function tends to one.To appear in
J. Geom. Anal. , 2016.[Kle78] Paul F. Klembeck. K¨ahler metrics of negative curvature, the Bergmann metric near theboundary, and the Kobayashi metric on smooth bounded strictly pseudoconvex sets.
Indiana Univ. Math. J. , 27(2):275–282, 1978.[KN96] Shoshichi Kobayashi and Katsumi Nomizu.
Foundations of differential geometry. Vol. II .Wiley Classics Library. John Wiley & Sons, Inc., New York, 1996. Reprint of the 1969original, A Wiley-Interscience Publication.[Kob05] Shoshichi Kobayashi.
Hyperbolic manifolds and holomorphic mappings. An introduction.2nd ed.
J. Differential Geom. , 68(3):571–637, 2004.[Roy71] H. L. Royden. Remarks on the Kobayashi metric. In
Several complex variables, II (Proc.Internat. Conf., Univ. Maryland, College Park, Md., 1970) , pages 125–137. LectureNotes in Math., Vol. 185. Springer, Berlin, 1971.[Ven89] Sergio Venturini. Pseudodistances and pseudometrics on real and complex manifolds.
Ann. Mat. Pura Appl. (4) , 154:385–402, 1989.[Yeu09] Sai-Kee Yeung. Geometry of domains with the uniform squeezing property.
Adv. Math. ,221(2):547–569, 2009.[Zim16] A. Zimmer. A gap theorem for the complex geometry of convex domains. To appear in