A new Schwarz-Pick Lemma at the boundary and rigidity of holomorphic maps
aa r X i v : . [ m a t h . C V ] M a r A NEW SCHWARZ-PICK LEMMA AT THE BOUNDARY AND RIGIDITY OFHOLOMORPHIC MAPS
FILIPPO BRACCI † , DANIELA KRAUS, AND OLIVER ROTHA BSTRACT . In this paper we establish several invariant boundary versions of the (infinitesi-mal) Schwarz-Pick lemma for conformal pseudometrics on the unit disk and for holomorphicselfmaps of strongly convex domains in C N in the spirit of the boundary Schwarz lemma ofBurns–Krantz. Firstly, we focus on the case of the unit disk and prove a general boundaryrigidity theorem for conformal pseudometrics with variable curvature. In its simplest cases thisresult already includes new types of boundary versions of the lemmas of Schwarz–Pick, Ahlfors–Schwarz and Nehari–Schwarz. The proof is based on a new Harnack–type inequality as well as aboundary Hopf lemma for conformal pseudometrics which extend earlier interior rigidity resultsof Golusin, Heins, Beardon, Minda and others. Secondly, we prove similar rigidity theoremsfor sequences of conformal pseudometrics, which even in the interior case appear to be new.For instance, a first sequential version of the strong form of Ahlfors’ lemma is obtained. Asan auxiliary tool we establish a Hurwitz–type result about preservation of zeros of sequences ofconformal pseudometrics. Thirdly, we apply the one–dimensional sequential boundary rigidityresults together with a variety of techniques from several complex variables to prove a boundaryversion of the Schwarz–Pick lemma for holomorphic maps of strongly convex domains in C N for N > C ONTENTS
1. Introduction 22. The Schwarz–Pick, Ahlfors–Schwarz and Nehari–Schwarz lemmas at the boundaryof the unit disk and in higher dimension 43. Rigidity of conformal pseudometrics 84. Rigidity of sequences of conformal pseudometrics 105. Proof of Theorem 3.2: A Harnack inequality for conformal pseudometrics 136. Proof of Theorem 4.4: Hurwitz’s theorem for conformal pseudometrics 207. Infinitesimal rigidity in strongly convex domains 278. Appendix: Theorem 2.1 vs. Burns–Krantz 34References 35 † Partially supported by PRIN Real and Complex Manifolds: Topology, Geometry and holomorphicdynam-ics n.2017JZ2SW5, by INdAM and by the MIUR Excellence Department Project awarded to the Department ofMathematics, University of Rome Tor Vergata, CUP E83C18000100006.
1. I
NTRODUCTION
The Schwarz lemma is one of the most basic results in complex analysis that capture therigidity of holomorphic mappings. Its uniqueness part or “strong form” guarantees that a holo-morphic selfmap of the open euclidean unit ball B N of C N which agrees with the identity map-ping up to order 1 at some interior point z is in fact the identity map (The historical sources areSchwarz [46, Vol. II, p. 110] for N = N > z is aboundary point. We mention the pioneering work of Loewner 1923 ([36, Hilfssatz I]) for thecase N = λ D ( z ) | dz | : = | dz | − | z | of the open unit disk D : = B among all conformal Riemannian pseudometrics λ ( z ) | dz | on D : if κ λ ( z ) denotes the Gauss curvature of λ ( z ) | dz | , then κ λ ( z ) ≤ κ λ D ( z ) = − z ∈ D implies(1.1) λ ( z ) ≤ λ D ( z ) for all z ∈ D . The Ahlfors lemma has been generalized in many directions, notably by Nehari [40] and Yau[53]. By analogy with the Schwarz lemma there is a strong form of the Ahlfors–Schwarzlemma: If equality holds in (1.1) at some point z ∈ D , then λ ( z ) = λ D ( z ) for any z ∈ D . Thisstriking interior rigidity property has first been established by Heins [25] in 1962, who usedit to investigate the value of the Bloch constant, the determination of which is one of the mostintriguing open problems in complex analysis of one variable. Later, different proofs of the caseof interior equality of Ahlfors’ lemma have been given by Royden [44] and Minda [39]. Onlyrecently, a corresponding interior rigidity result for (a generalization of) Nehari’s extension ofthe Ahlfors–Schwarz lemma has been obtained in [30].The main purpose of the present paper is to prove invariant boundary rigidity results forholomorphic selfmaps of the unit disk and of strongly convex domains in C N as well as forconformal pseudometrics on the disk in the spirit of the boundary Schwarz lemma of Burns andKrantz. This includes boundary rigidity versions of(a) the Schwarz–Pick lemma for holomorphic selfmaps of B N for any N ≥
1, and(b) the Ahlfors–Schwarz as well as the Nehari lemma for conformal pseudometrics in D . IGIDITY 3
For the case N = sequences of such metrics. To the best of our knowledge these results are new even in the “interior”case. They include, in particular, sequential versions of the strong Schwarz–Pick, the strongAhlfors–Schwarz and the strong Nehari–Schwarz lemma. The sequential version of the one–dimensional Schwarz–Pick lemma at the boundary will play a decisive role for the proof of theSchwarz–Pick lemma for holomorphic selfmaps of the unit ball in C N at the boundary. In fact,this latter result has been one of our primary sources of motivation for looking at rigidity resultsfor sequences of conformal pseudometrics. Along the way we prove a new Hurwitz–type resultfor preservation of zeros of conformal pseudometrics (Theorem 6.1).As we will see, the results of this paper contain and extend many of the existing interiorand boundary rigidity results for holomorphic maps and conformal metrics. In particular, theboundary Schwarz–Pick lemma for the unit disk is easily seen to imply the one–dimensionalboundary Schwarz lemma of Burns and Krantz. However, unlike the Burns–Krantz theoren,the boundary Schwarz–Pick lemma – just as the other results of this paper – is conformallyinvariant , so it easily extends to holomorphic from the disk into hyperbolic Riemann surfaces.The organization of this paper is as follows. In an introductory Section 2 we state and discussboundary versions of the Lemmas of Schwarz–Pick for the disk, the ball and strongly convexdomains, of the Ahlfors–Schwarz lemma and of (an extension of) the Nehari–Schwarz lemma.This is done primarly for the sake of readability as it allows to easily grasp the essential core ofour work at a fairly nontechnical level and also to put the results into context with prior work.In Sections 3 and 4 we describe two general boundary rigidity results for conformal metricswith variable negative curvature on the unit disk, which contain (most of) the results of Section2 as special cases. Section 4 concludes with a sequential version of the boundary Schwarz–Pick lemma of Section 2 which is needed for extensions to higher dimensions. In Sections 5we discuss and prove Theorem 3.2, one of the main contributions of this paper. It providesa Harnack–type inequality as well as a corresponding boundary Hopf lemma for conformalpseudometrics. Section 6 presents a Hurwitz–type result for conformal pseudometrics which isthen combined with the Harnack inequality of Theorem 3.2 to prove our main rigidity result forsequences of conformal pseudometrics on the unit disk. Then attention shifts to several complexvariables. In Section 7 we state and prove a boundary rigidity result of Schwarz–Pick type forholomorphic maps of strongly convex domains. F. BRACCI, D. KRAUS, AND O. ROTH
2. T HE S CHWARZ –P ICK , A
HLFORS –S CHWARZ AND N EHARI –S CHWARZ LEMMAS AT THEBOUNDARY OF THE UNIT DISK AND IN HIGHER DIMENSION
The Schwarz–Pick lemma at the boundary.
The classical Schwarz–Pick lemma saysthat if f is a holomorphic selfmap of the open unit disk D = { z ∈ C : | z | < } , then f h ( z ) : = (cid:0) − | z | (cid:1) | f ′ ( z ) | − | f ( z ) | ≤ z ∈ D . The “strong” form of the Schwarz-Pick lemma guarantees that f h ( z ) = z ∈ D if and only if f belongs to the group Aut ( D ) of conformal automorphisms of D . This “interior rigidity” result has the following counterpart on the boundary, which can beproved in various ways (see e.g. Remark 5.6 below). Its main purpose for us is to provide theprime motivating example throughout this paper. Theorem 2.1 (The strong form of the Schwarz–Pick lemma at the boundary of the unit disk) . Suppose f is a holomorphic selfmap of D such that (2.1) f h ( z n ) = + o (cid:0) ( − | z n | ) (cid:1) for some sequence ( z n ) in D with | z n | → . Then f ∈ Aut ( D ) . The error term is sharp. For f ( z ) = z we have f h ( z ) = | z | + | z | = − ( − | z | ) + | z | = − ( − | z | ) + o (cid:0) ( − | z | ) (cid:1) ( | z | → ) . Hence one cannot replace “little o ” by “big O ” in Theorem 2.1. Example 2.8 below will showthat Theorem 2.1 is sharp even for univalent selfmaps of the unit disk. Remark . From Theorem 2.1 one caneasily deduce the well–known boundary Schwarz lemma of Burns and Krantz [14], which as-serts that if f is a holomorphic selfmap of D such that(2.2) f ( z ) = z + o (cid:0) | − z | (cid:1) as z → , then f ( z ) ≡ z . In fact, a fairly straightforward application of Cauchy’s integral formula for f ′ shows that if (2.2) holds nontangentially, then f h ( z ) = + o (cid:0) | − z | (cid:1) as z → , see Section 8 for the details. Hence Theorem 2.1 gives f ∈ Aut ( D ) , and it now follows easilythat (2.2) implies f ( z ) ≡ z .In order to state the generalization of Theorem 2.1 to higher dimension, we first remark thatfor z ∈ D , f h ( z ) = k D ( f ( z ) ; f ′ ( z )) k D ( z ; 1 ) , where k D ( z ; v ) = v −| z | denotes the infinitesimal hyperbolic (Poincar´e) metric of D . IGIDITY 5
Then, condition (2.1) can be rewritten as k D ( f ( z n ) ; f ′ ( z n )) = k D ( z n ; 1 ) + k D ( z n ; 1 ) o ( δ D ( z n )) , where, if D ⊂ C N is a domain and w ∈ D , we let δ D ( w ) : = inf {| z − w | : z ∈ C N \ D } . Taking into account that k D ( z n ; 1 ) ∼ / δ D ( z n ) , it follows that (2.1) is equivalent to k D ( f ( z n ) ; f ′ ( z n )) = k D ( z n ; 1 ) + o ( δ D ( z n )) . This is the condition that can be generalized to higher dimension replacing k D with the (infini-tesimal) Kobayashi metric k D of a domain D ⊂ C N .We state here the result for holomorphic selfmaps of the unit ball B N : = { z ∈ C N : | z | < } of C N w.r.t. the euclidean norm | z | : = ( | z | + . . . + | z N | ) / for z = ( z , . . . , z N ) ∈ C N .Recall that, the complex tangent space T C p ∂ B N of ∂ B N at p ∈ ∂ B N is defined as T C p ∂ B N : = { v ∈ C N : h v , p i = } , where h v , p i = ∑ Nj = v j p j is the standard Hermitian product in C N . Also, we denote by Π p ( v ) the orthogonal projection of v on T C p ∂ B N . Finally, if z ∈ B N is sufficiently close to the boundary,we let π ( z ) ∈ ∂ B N be the closest point. With this notation, we can state the following: Theorem 2.3 (The strong form of the Schwarz–Pick lemma at the boundary of the unit ball) . Let F be a holomorphic selfmap of the open unit ball B N of C N . Then F is a biholomorphism ifand only if(1) the cluster set of { F ( z n ) } belongs to ∂ B N for every { z n } ⊂ B N converging to e and suchthat { ( z n − e ) / | z n − e |} converges to some τ ∈ T C e ∂ B N (or, equivalently, lim n → ∞ ( z n , − e ) / | z n − e | = );(2) for every v ∈ C N , | v | = , v = a) if { z k } ⊂ B N ∩ ( C v + e ) is a sequence converging to e non-tangentially such that { F ( z k ) } has no accumulation points in B N then lim sup k → ∞ | Π π ( F ( z k )) ( dF z k ( v )) | < + ∞ , b) and for ( B N ∩ ( C v + e )) ∋ z → e non-tangentially,k B N ( F ( z ) ; dF z ( v )) = k B N ( z ; v ) + o ( δ B N ( z )) . Some remarks about this statement are in order. First, note that if N = T C e ∂ D = { } ,hence hypothesis (1) and hypothesis (2).a) hold trivially, and Theorem 2.3 reduces to (a versionof) Theorem 2.1. These extra hypotheses reflect the fact that in higher dimension there is ingeneral little control on the complex tangential directions. However, we presently do not knowif such hypotheses are really necessary.If F is known to be continuous at p and F ( p ) = q ∈ ∂ B N , hypothesis (1) is automaticallysatisfied and (2).a) can be replaced by assuming | Π q ( dF z k ( v ) | ≤ C < + ∞ . F. BRACCI, D. KRAUS, AND O. ROTH
One can replace hypothesis (2).b) by assuming that for every v there exists a sequence { z n } contained in ( B N ∩ ( C v + e )) and converging non-tangentially to e for which the conclu-sion holds. Theorem 2.3 (whose proof is contained in Section 7) is a special case of Theo-rem 7.5, where the result is extended to holomorphic maps between strongly convex domainswith smooth boundary.One can replace hypotheses (1) and (2).a) by assuming a boundedness condition on dF z as z → e . We state the result for general strongly convex domains (the proof is given in Section 7): Theorem 2.4.
Let D , D ′ ⊂ C N be two bounded strongly convex domains with smooth boundary,let F : D → D ′ be holomorphic, and let p ∈ ∂ D. Then F is a biholomorphism if and only if(1) there exists C > such that lim sup z → p | dF z ( v ) | ≤ C for all v ∈ C N with | v | = ,(2) and k D ′ ( F ( z ) ; dF z ( w )) = k D ( z ; w ) + o ( δ D ( z )) , when z → p non-tangentially and locally uniformly in w ∈ C N \ T C p ∂ D, | w | = . As a direct consequence (just apply the previous result to F ( z ) = z ) we have the followinginteresting fact: Corollary 2.5.
Let D , D ′ ⊂ C N be two bounded strongly convex domains with smooth boundary.Assume that D ⊆ D ′ . If there exists p ∈ ∂ D such thatk D ′ ( z ; v ) = k D ( z ; v ) + o ( δ D ( z )) , for z → p non-tangentially, locally uniformly in v ∈ C N \ T C p ∂ D, | v | = , then D = D ′ . The Ahlfors lemma at the boundary.
In contrast to the Burns–Krantz result, Theorem2.1 is manifestly a conformally invariant statement and is therefore perhaps best understood asa special case of the following boundary version of the Ahlfors–Schwarz lemma [3].
Theorem 2.6 (The strong form of the Ahlfors–Schwarz lemma at the boundary) . Let λ ( z ) | dz | be a conformal pseudometric on D with curvature κ λ ≤ − . Suppose that λ ( z n ) λ D ( z n ) = + o (cid:0) ( − | z n | ) (cid:1) for some sequence ( z n ) in D such that | z n | → . Then λ = λ D . By a conformal pseudometric λ ( z ) | dz | on a domain G in C we mean a continuous nonnega-tive function λ : G → R such that the curvature(2.3) κ λ ( z ) = − ∆ ( log λ )( z ) λ ( z ) of λ ( z ) | dz | is defined for all z ∈ G with λ ( z ) >
0. For simplicity, we assume throughout thispaper that λ is of class C in { z ∈ G : λ ( z ) > } , so ∆ in (2.3) is the standard Laplacian. IGIDITY 7
Expressions such as “ κ ≤ −
4” will always mean that κ ( z ) ≤ − z ∈ G with λ ( z ) >
0. If λ is strictly positive on G we call λ ( z ) | dz | conformal metric. Note that the Gauss curvature ofthe Poincar´e metric λ D ( z ) | dz | is constant −
4. Hence, Theorem 2.1 follows from Theorem 2.6applied to the conformal pseudometric λ ( z ) | dz | : = ( f ∗ λ D )( z ) | dz | : = | f ′ ( z ) | − | f ( z ) | | dz | , by noting κ λ = − f h ( z ) = λ ( z ) / λ D ( z ) = f ∈ Aut ( D ) . Remark . Recall that the Ahlfors–Schwarz lemma [3] says that λ ( z ) ≤ λ D ( z ) for all z ∈ D and for anyconformal pseudometric λ ( z ) | dz | on D with curvature bounded above by −
4. The “strong”form of the Ahlfors–Schwarz lemma ([39]) guarantees that if in addition λ ( z ) = λ D ( z ) at some interior point z ∈ D , then λ ≡ λ D , see Jørgensen [26] and Heins [25] as well as [44, 39, 18] fordifferent proofs. Hence Theorem 2.6 can be viewed as a boundary version of the strong form ofthe Ahlfors–Schwarz lemma. Example 2.8.
For ε > f ε ( z ) : = z − ε ( z − ) . Then f ε ( D ) ⊂ D if and only if ε ≤ /
4, and f ε is a (locally) univalent selfmap of D if and only if ε ≤ /
12. Hence λ ε ( z ) : = | f ′ ε ( z ) | − | f ε ( z ) | defines a conformal metric for 0 < ε ≤ /
12 and a conformal pseudometric for 1 / < ε ≤ / D such that κ λ ε ≡ −
4. In addition, a computation shows λ ε ( z ) λ D ( z ) = f h ε ( z ) = − ε ( − | z | ) + o (cid:0) ( − | z | ) (cid:1) as z → . This implies that Theorem 2.6 as well as Theorem 2.1 are best possible even for conformal metrics and univalent selfmaps of D .Theorem 2.6 is “ready to use” for establishing in a standard way a general boundary rigidityresult for holomorphic maps from the disk into hyperbolic Riemann surfaces. Corollary 2.9.
Let R be a Riemann surface with the unit disk as universal covering surface andlet µ ( z ) | dz | be a conformal pseudometric on R with curvature κ µ ≤ − . Suppose that f : D → Ris a holomorphic mapping such that µ ( f ( z n )) | f ′ ( z n ) | λ D ( z n ) = + o (cid:0) ( − | z n | ) (cid:1) for some sequence ( z n ) in D with | z n | → . Then f : D → R is a universal covering map of R(and µ ( z ) | dz | = λ R ( z ) | dz | , the Poincar´e metric of R). F. BRACCI, D. KRAUS, AND O. ROTH
The Nehari–Schwarz lemma at the boundary.
The assumptions in Theorem 2.1 andCorollary 2.9 rule out that the map f has critical points and in particular that it is a branchedcovering. In order to incorporate branching, we recall Nehari’s sharpening [40] of the Schwarz–Pick lemma: If f is a holomorphic selfmap of D and B is a finite Blaschke product such thatf ′ / B ′ has a holomorphic extenison to D ‡ , then f h ( z ) ≤ B h ( z ) for all z ∈ D . If, in addition,f h ( z ) / B h ( z ) = for some z ∈ D , then f = T ◦ B for some T ∈ Aut ( D ) , that is, f is a fi-nite Blaschke product. Nehari’s result has been generalized in [28, 30] as follows:
For everynonconstant holomorphic selfmap f of the unit disk there is always a (not necessarily finite)Blaschke product B such that f h ( z ) ≤ B h ( z ) for all z ∈ D and such that f ′ and B ′ have thesame zeros counting multiplicities. Such Blaschke products only depend on their critical points(counting multiplicities) and are uniquely determined up to postcomposition with a unit diskautomorphism; they are called maximal Blaschke products. Moreover, if f h ( z ) / B h ( z ) = forsome z ∈ D § then f = T ◦ B for some T ∈ Aut ( D ) . Since finite Blaschke products are exactlythe maximal Blaschke products corresponding to finitely many critical points (of B ), see again[28, 30], Nehari’s generalization of Schwarz’ lemma is just the special case dealing with finitelymany critical points. The following theorem handles the case of equality at the boundary in thepresence of branch points. Theorem 2.10 (The strong form of the generalized Nehari–Schwarz lemma at the boundary) . Let f be a holomorphic selfmap of D and B a maximal Blaschke product such that f ′ / B ′ has aholomorphic extension to D . If f h ( z n ) B h ( z n ) = + o (cid:0) ( − | z n | ) (cid:1) for some sequence ( z n ) in D such that | z n | → , then f = T ◦ B for some T ∈ Aut ( D ) and f is amaximal Blaschke product. In particular, if the Blaschke product B in Theorem 2.10 is a finite Blaschke product, then f is also a finite Blaschke product (with the same critical points as B ). This provides a boundaryversion of the case of equality in Nehari’s generalization of the Schwarz–Pick lemma. The case B = id is exactly the boundary Schwarz–Pick lemma, so Theorem 2.10 generalizes Theorem2.1. 3. R IGIDITY OF CONFORMAL PSEUDOMETRICS
In this section we state one of the main results of the present paper, a boundary rigiditytheorem for conformal pseudometrics. It includes all the one–dimensional results of Section 2as special cases.It is convenient to introduce the following terminology. ‡ . Or equivalently, if ξ ∈ D is a zero of B ′ of order k , then ξ is a zero of f ′ of order at least k . § . If z is a critical point of B , then the assumption “ f h ( z ) B h ( z ) =
1” has to be interpreted as lim z → z f h ( z ) B h ( z ) = IGIDITY 9
Definition 3.1.
Let λ ( z ) | dz | and µ ( z ) | dz | be conformal pseudometrics defined on a domain G in C . We say that λ is dominated by µ and write λ (cid:22) µ whenever(i) κ λ ≤ κ µ in G , and(ii) µ ( z ) | dz | has only isolated zeros in G and λ / µ has a continuous extension from { z ∈ G : µ ( z ) = } to G with values in [ , ] .If λ (cid:22) µ , we write λ ( z ) / µ ( z ) for the value of the continuous extension at any point z ∈ G .If f and g are holomorphic selfmaps of D and f is not constant, then it is clear that g h ≤ f h if and only if g ∗ λ D (cid:22) f ∗ λ D .We can now state our main boundary rigidity result for conformal pseudometrics. Theorem 3.2 (Boundary ridigity for conformal pseudometrics with variable negative curvature) . Let λ ( z ) | dz | and µ ( z ) | dz | be conformal pseudometrics on D such that κ µ = κ for some locallyH¨older continuous function κ : D → R with − c ≤ κ ( z ) ≤ − in D for some c > . Suppose λ (cid:22) µ and (3.1) λ ( z n ) µ ( z n ) = + o (cid:16) ( − | z n | ) c / (cid:17) for some sequence ( z n ) in D such that | z n | → . Then λ = µ . Remark . If the conformal pseudometric µ ( z ) | dz | in Theorem 3.2 is a conformal metric , thatis, µ is zerofree, then it suffices to assume that κ µ = κ for some continuous function κ : D → R with − c ≤ κ ( z ) ≤ − D for some c >
0. This will follow from the proof of Theorem 3.2.The H¨older–condition in Theorem 3.2 is a compromise between generality and readability. Itguarantees that we can always consider classical C –solutions of the corresponding curvatureequation, apply the usual maximum principle etc.. This assumption is sufficiently general forall applications in the present paper, so we restrict ourselves to this case here and do not strivefor highest generality with respect to regularity assumptions. Remark . We do not need to assume a lower bound for the curvature of the pseudometric λ ( z ) | dz | in Theorem 3.2. This is in sharp contrast to Theorem 4.4 below which deals with avariant of Theorem 3.2 for sequences of conformal pseudometrics. Remark . Theorem 3.2 includes(a) the strong form of the Ahlfors–Schwarz lemma at the boundary (Theorem 2.6): this isthe special case µ = λ D of Theorem 3.2, since κ λ ≤ − λ (cid:22) λ D .(b) the strong form of the generalized Nehari–Schwarz lemma at the boundary (Theorem2.10): this is the special case µ = B ∗ λ D where B is a maximal Blaschke product and λ = f ∗ λ D for some holomorphic selfmap f of D such that f ′ / B ′ has a holomorphic extensionto D , so [30, Corollary 3.1] shows that f h ≤ B h and hence λ (cid:22) µ . Remark . We wish to point out that if we assume in Theorem 3.2 that condition (3.1) holds fora sequence ( z n ) in D tending to some interior point z ∈ D , then the proof below will show thatwe also have λ = µ , even under the milder assumption that κ is only bounded from above by −
4. We thereby obtain an extension of the main result in [30, Theorem 2.2] which has coveredthe constant curvature case κ ≡ −
4. The main new ingredient of Theorem 3.2, however, is thefact that the “asymptotic” equality (3.1) for some boundary sequence ( z n ) already forces λ = µ . Remark . The condition − c ≤ κ µ ≤ − c > C N or complex manifolds,see e.g. [10]. It often goes by the name “negatively pinched”. We see that the pinching constant c controls the error term in the asymptotic relation (3.1).It turns out that the behaviour of a negatively pinched conformal pseudometric µ ( z ) | dz | at itsisolated zeros can be precisely described: Lemma 3.8 (Isolated zeros of negatively pinched conformal pseudometrics, [29, Theorem 3.2]) . Let µ ( z ) | dz | be a conformal pseudometric on the disk K r ( ξ ) = { z ∈ C : | z − ξ | < r } withcenter ξ ∈ C and radius r > such that µ ( ξ ) = and µ > on K r ( ξ ) \ { ξ } . Suppose that − c ≤ κ µ ≤ − for some constant c < . Then there is unqiue real number α > such that thelimit lim z → ξ µ ( z ) | z − ξ | α exists and is = . The number α > order of the isolated zero ξ of the pseudometric µ ( z ) | dz | .We understand that, whenever we speak of the order of an isolated zero of a conformal pseudo-metric µ ( z ) | dz | , we always implicitly assume that µ ( z ) | dz | is negatively pinched in a punc-tured neighborhood of this zero as in Lemma 3.8. Occasionally, it will be convenient toslightly abuse language and call ξ a zero of µ ( z ) | dz | of order α = µ ( ξ ) =
0. Note thatif µ ( z ) | dz | = f ∗ λ D ( z ) | dz | for some nonconstant holomorphic selfmap f of D , then µ has a zero ξ of order α > ξ is a branch point of f of order α >
0, that is, a zero of f ′ order α >
0. 4. R
IGIDITY OF SEQUENCES OF CONFORMAL PSEUDOMETRICS
We now turn to rigidity results for sequences of conformal pseudometrics. These results willalso be needed for the proof of Theorem 2.3. We are dealing with problems of the followingtype:
Problem . Let ( λ n ) be asequence of conformal pseudometrics such that κ λ n ≤ − D . Suppose that either(i) (Interior case) λ n ( z ) → λ D ( z ) for some interior point z , or IGIDITY 11 (ii) (Boundary case) λ n ( z n ) λ D ( z n ) = + o (cid:0) ( − | z n | ) (cid:1) for some sequence ( z n ) in D tending to ∂ D .Does it follow that λ n → λ D locally uniformly in D ?By the strong form of the Ahlfors–Schwarz lemma at the boundary for a single conformalpseudometric (cf. Remark 2.7 for the interior case and Theorem 2.6 for the boundary case) theanswer is “yes” if λ = λ = . . . . However, in general, the answer is “No” in both cases! Thefollowing two examples illustrate that there are at least two phenomena which play a role here. Example 4.2.
Consider s n ( z ) : = − − n ! + (cid:18) | z | + n ! (cid:19) / n , n ∈ N , z ∈ D . Then each s n is a negative smooth subharmonic function in D and therefore λ n ( z ) | dz | = e s n ( z ) λ D ( z ) | dz | is a conformal metric on D such that κ λ n ≤ −
4. Note that s n ( z n ) → ( z n ) in D which is bounded way from the origin. In fact, one can show that for each N ∈ N there is asequence ( z n ) in D such that | z n | → λ n ( z n ) λ D ( z n ) = + o (cid:0) ( − | z n | ) N (cid:1) . However, λ n ( ) → λ D ( ) / e , whereas λ n → λ D locally uniformly in D \ { } . Note that κ λ n ( ) →− ∞ . Example 4.3.
Let ( α n ) be a sequence of positive real numbers α n < α n →
0. Then µ n ( z ) | dz | : = ( + α n ) | z | α n − | z | ( + α n ) | dz | are conformal pseudometrics on D such that κ µ n ≡ −
4. Note that µ n ( z ) → λ D ( z ) for any z ∈ D \ { } whereas µ n ( ) = n = , , . . . . Now we choose z n ∈ D \ { } such that z n → | z n | α n →
1, and consider the unit disk automorphisms T n ( z ) : = z n − z − z n z . Then λ n ( z ) | dz | : = T ∗ n µ n ( z ) | dz | are conformal pseudometrics in D with κ λ n ≡ −
4. Moreover, λ n ( ) = ( + α n ) | z n | α n − | z n | ( + α n ) (cid:0) − | z n | (cid:1) → = λ D ( ) . We note that λ n → λ D pointwise , but not locally uniformly in D . These examples show in particular that there is no full sequential version of the strong formof the Ahlfors–Schwarz lemma. Note that in Example 4.2 there is no (locally uniform) lowerbound on the curvatures, while in Example 4.3 the zeros of ( λ n ) “disappear” as n → ∞ . Thefollowing result shows that a certain control over the curvature as well as the potential zerosdoes give a version of the strong Ahlfors–Schwarz lemma for sequences. This is our mainresult for sequences of conformal pseudometrics. Theorem 4.4 (Boundary and interior rigidity for sequences of conformal pseudometrics) . Let λ n ( z ) | dz | , n = , , . . . , be conformal pseudometrics on D with only isolated zeros so that ( κ λ n ) is uniformly bounded below in D . Let µ ( z ) | dz | be a conformal pseudometric with κ µ = κ forsome locally H¨older continuous function κ : D → R with − c ≤ κ ( z ) ≤ − for some c > .Suppose λ n (cid:22) µ for any n = , , . . . and (4.1) λ n ( z n ) µ ( z n ) = + o (cid:16) ( − | z n | ) c / (cid:17) for some sequence ( z n ) in D which either is compactly contained in D (interior case) or else | z n | → (boundary case). Then there are the following alternatives: either(i) λ n / µ → locally uniformly on D , or(ii) there is a point ξ ∈ D and a subsequence ( λ n k ) such that every λ n k has a zero ξ n k ∈ D of order α n k > such that ξ n k → ξ and α n k → . Condition (ii) in Theorem 4.4 roughly says that there is an accumulation of zeros which fadeaway as k → ∞ . In the interior case of Theorem 4.4 we shall see that it suffices to assume that ( κ λ n ) is locally uniformly bounded below in D . Remark . Choosing µ = λ D and aconstant sequence ( z n ) = ( z ) for some fixed point z ∈ D , condition (4.1) just means that λ n ( z ) → λ D ( z ) , so this special case of Theorem 4.4 gives a solution to Problem 4.1 for theinterior case. If ( z n ) tends to ∂ D , then Theorem 4.4 gives a solution to Problem 4.1 for theboundary case. As illustrated by Examples 4.2 and 4.3, Theorem 4.4 is essentially best possi-ble.Under the conditions of Theorem 4.4, we see that if ( λ n ) is a sequence of metrics (i.e. noneof the λ n ’s has a zero) or if there is a constant α > λ n has only zeros of orderat least α , then λ n → µ locally uniformly in D . This applies in particular to the case µ ( z ) | dz | = λ D ( z ) | dz | , λ n ( z ) | dz | = f ∗ n λ D ( z ) | dz | for holomorphic maps f n : D → D , since any λ n has only zeros of order at least 1. We there-for arrive at the following full sequential version of the Schwarz–Pick lemma at the boundary(Theorem 2.1). Corollary 4.6 (The sequential Schwarz–Pick lemma at the boundary) . Let ( f n ) be a sequenceof holomorphic maps f n : D → D such thatf hn ( z n ) = + o (cid:0) ( − | z n | ) (cid:1) IGIDITY 13 for some sequence ( z n ) in D with | z n | → . Then(a) f hn → locally uniformly in D , and(b) every locally uniformly subsequential limit of ( f n ) is either a unit disk automorphism ora unimodular constant.Remark . Choosing µ ( z ) | dz | as the pull-back of λ D ( z ) | dz | under some finite (or maximal) Blaschke product in Theorem 4.4 we obtain ina similar way a sequential version of the Nehari–Schwarz lemma. We leave it to the interestedreader to provide the corresponding statement and proof.5. P ROOF OF T HEOREM
ARNACK INEQUALITY FOR CONFORMALPSEUDOMETRICS
The main ingredient for the proofs of the results in Section 3 is the following Harnack–typeestimate for negatively curved conformal pseudometrics.
Theorem 5.1 (Boundary Harnack inequality and higher–order Hopf lemma for pseudometrics) . For any r ∈ ( , ) there is a universal constant C r > with the following property. Let λ ( z ) | dz | and µ ( z ) | dz | be conformal pseudometrics on D with κ µ = κ for some locally H¨older continuousfunction κ : D → R such that − c ≤ κ ( z ) ≤ − for some c > . Suppose that λ (cid:22) µ . Then (5.1) log λ ( z ) µ ( z ) ≤ C r ( − r ) c / · (cid:18) max | ξ | = r log λ ( ξ ) µ ( ξ ) (cid:19) (cid:0) − | z | (cid:1) c / on r ≤ | z | < . In particular, if λ ( z ) / µ ( z ) < for one point z ∈ D , then λ ( z ) / µ ( z ) < for all z ∈ D and (5.2) lim sup | z |→ log λ ( z ) µ ( z ) ( − | z | ) c / < . Remark . Inequality (5.1) is a Harnack–type estimate for conformal pseudometrics “up tothe boundary”. (5.2) might be viewed as boundary Hopf–lemma of higher order.
Remark . The proof will show that we can take for instance(5.3) C r = e − / r . The proof of Theorem 5.1 will also show that if µ ( z ) | dz | is a conformal metric , so µ > D , then the statement of Theorem 5.1 stays valid if κ : D → R is merely continuousinstead of being locally H¨older continuous.A straightforward consequence of Theorem 5.1 is the following result. Corollary 5.4.
Let κ : D → R be a locally H¨older continuous function with κ ( z ) ≤ − for allz ∈ D and let < r < R < . Then there is a positive constant C = C ( r , R , κ ) such that (5.4) λ ( z ) µ ( z ) ≤ max | ξ | = r (cid:18) λ ( ξ ) µ ( ξ ) (cid:19) C on r ≤ | z | ≤ R for all conformal pseudometrics λ ( z ) | dz | and µ ( z ) | dz | with κ µ = κ and λ (cid:22) µ . In particular,if λ ( z ) / µ ( z ) < for one point z ∈ D , then λ ( z ) / µ ( z ) < for every z ∈ D .Remark . Under the assumptions of Corollary 5.4 the conclusion that λ ( z ) / µ ( z ) = z ∈ D implies λ = µ everywhere has been proved before in [30, Theorem 2.2] in thespecial case κ = −
4. This has provided the main step for handling the case of interior equalityin the general Nehari–Schwarz lemma. The main new aspect of Corollary 5.4 is twofold. First,it extends the main result of [30] to the case of variable (strictly negative) curvature. Second,it provides the quantitative bound (5.4), which is essential for proving the sequential version ofthe strong form of Nehari–Schwarz lemma. For such purposes, good control of the “Harnackconstant” C ( r , R , κ ) will be essential. In fact, the proof will show that the Harnack constant C ( r , R , κ ) in Corollary 5.4 can be chosen as C ( r , R , κ ) : = exp (cid:18) − ρ r (cid:19) (cid:18) ρ − R ρ − r (cid:19) c ρ ( κ ) / , c ρ ( κ ) : = − min | z |≤ ρ κ ( z ) > , for some fixed ρ ∈ ( R , ) . For later purpose we note that this Harnack constant is monotonicallydecreasing w.r.t. c ρ ( κ ) . Proof of Theorem 5.1.
The proof is divided into several steps.(i) We consider u ( z ) : = log λ ( z ) , u ( z ) : = log µ ( z ) and u ( z ) : = u ( z ) − u ( z ) = log λ ( z ) µ ( z ) ≤ z ∈ D . Let D : = D \ { z ∈ D : λ ( z ) = } . Since ∆ u = − κ λ ( z ) e u ≥ − κ ( z ) e u on D and ∆ u = − κ ( z ) e u on D , we have ∆ u ≥ − κ ( z ) (cid:0) e u − e u (cid:1) = − κ ( z ) e u (cid:0) e u − (cid:1) on D . Using the elementary inequality e y − ≥ y , which is valid for all y ∈ R , together with κ ( z ) ≤ ∆ u ≥ − κ ( z ) e u u on D . Since e u ( z ) = µ ( z ) ≤ λ D ( z ) = − | z | by the Ahlfors–Schwarz lemma, we deduce from κ ( z ) ≥ − c and u ≤ ∆ u ≥ c ( − | z | ) u on D . (ii) We now fix r ∈ ( , ) and show that v r ( z ) = ( − | z | ) c / e ( −| z | ) / r is an explicit solution of the partial differential inequality (5.5) on the annulus r ≤ | z | <
1. Inorder to prove(5.6) ∆ v r ≥ c ( − | z | ) v r for all r ≤ | z | < , we first observe ∆ v r ( z ) v r ( z ) ( − | z | ) = f ( | z | ) where f ( x ) : = x − ( + ( + c ) r ) x + ( + ( + c ) r + c r ) x − r ( + cr ) r . Now f ′ ( x ) = x − ( + ( + c ) r ) x + c r + ( + c ) r + r has a zero at the point x r : = + ( + c ) r + p + ( c − ) r + ( + c + c ) r . Since c ≥ r >
0, we see that x r ≥ + r + √ + r + r > . This means that the cubic polynomial f has its unique local minimum at the point x r >
1. Usingagain c ≥
4, we have f ( r ) = c + c ( c − ) r ≥ c and f ( ) = ( c − ) c ≥ c . Hence we get that f ( x ) ≥ c for all r ≤ x ≤ ε : = ε r : = − v r ( r ) · max | ξ | = r u ( ξ ) ≥ w r : = u + ε v r . By construction, w r ( z ) ≤ | z | = r and since v r ( z ) = | z | =
1, we also havelim sup | z |→ w r ( z ) = lim sup | z |→ u ( z ) ≤ . We claim that(5.7) w r ( z ) ≤ r ≤ | z | < . This gives u ( z )( − | z | ) c / = w r ( z )( − | z | ) c / − ε v r ( z )( − | z | ) c / ≤ − ε v r ( z )( − | z | ) c / = − ε e ( −| z | ) / r ≤ − ε for all r ≤ | z | <
1, which is the estimate (5.1).In order to prove (5.7), we assume that w r is positive somewhere in r ≤ | z | <
1. Then w r attains its maximal value w r ( z ) > r ≤ | z | ≤ z with r < | z | < z is not a zero of µ . Then z ∈ D and so0 ≥ ∆ w r ( z ) = ∆ u ( z ) + ε∆ v r ( z ) ≥ c ( − | z | ) w r ( z ) > , where we have used (5.5) and (5.6). This contradiction shows that (5.7) holds. In particular, if µ ( z ) | dz | is a conformal metric, the proof of (5.1) is finished, and we do not have used the localH¨older continuity of κ .(v) We now prove (5.7) when z is a zero of µ ( z ) | dz | of order α >
0, say. In view of λ (cid:22) µ ,the function λ / µ has a continuous extension to z . Since w r ( z ) >
0, we see thatlim z → z λ ( z ) µ ( z ) > . In particular, z is an isolated zero of λ ( z ) | dz | of order α , so there is an open disk K , which iscompactly contained in r < | z | < z ∈ K and λ > K \ { z } . We are confrontedwith the problem that we do not know a priori whether λ / µ has an C –extension to z . Inorder to deal with this problem we make use of the assumption that κ : D → R is locally H¨oldercontinuous. Then standard elliptic PDE theory shows that there is a unique continuous positivefunction v : K → R , which is of class C in K , such that ∆ log v = − κ ( z ) | z − z | α v ( z ) , z ∈ K , and v ( ξ ) = λ ( ξ ) | ξ − z | α , ξ ∈ ∂ K . We claim that(5.8) λ ( z ) | z − z | α ≤ v ( z ) ≤ µ ( z ) | z − z | α , z ∈ K \ { z } . IGIDITY 17
For the proof of the inequality on the right–hand side we note thatlog µ ( z ) | z − z | α , z ∈ K \ { z } , has a C –extension ˜ u : K → R by [29, Theorem 1.1] and therefore ∆ ˜ u ( z ) = − κ ( z ) | z − z | α e u ( z ) , z ∈ K . Hence ˜ u and log v are solutions to the same PDE. Since ˜ u ( ξ ) ≥ log v ( ξ ) for each ξ ∈ ∂ K , themaximum principle applied to this PDE easily shows that v ( z ) ≤ e ˜ u ( z ) for all z ∈ K . This provesthe right–hand side of (5.8). In order to prove the inequality on the left–hand side of (5.8), weconsider s ( z ) : = max (cid:26) , log λ ( z ) | z − z | α v ( z ) (cid:27) , z ∈ K \ { z } . If z ∈ K \ { z } such that s ( z ) >
0, then ∆ s ( z ) = ∆ log λ ( z ) − ∆ log v ( z ) ≥ − κ ( z ) (cid:2) λ ( z ) − | z − z | α v ( z ) (cid:3) ≥ , and s is subharmonic on K \ { z } . Moreover, s is bounded above at z sincelim sup z → z s ( z ) ≤ lim sup z → z log λ ( z ) | z − z | α v ( z ) < ∞ , which follows from the facts that λ ( z ) | dz | has a zero of order α at z and v ( z ) >
0. Therefore,the function s has a subharmonic extension to K with vanishing boundary values, so s ≤ w r ( z ) : = log v ( z ) − ˜ u ( z ) + ε v r ( z ) z ∈ K , and observe that ˜ w r is continuous on K and of class C in K . In addition, ˜ w r = w r on ∂ K as wellas ˜ w r ≥ w r in K in view of the left–hand side of (5.8). Since we have assumed that w r attainsits positive maximal value on K at z ∈ K , we see that ˜ w r attains its positive maximal value on K at some point z ∈ K . Now the same computation as in (iv) leads to0 ≥ ∆ ˜ w r ( z ) ≥ c ( − | z | ) ˜ w r ( z ) > , a contradiction. Hence (5.7) is proved also in the case that z is a zero of µ ( z ) | dz | , and we havecompletely proved (5.1).(vi) If λ ( z ) / µ ( z ) < z ∈ D , we put T ( z ) : = z − z − z z and consider ˜ µ ( z ) | dz | : = T ∗ µ ( z ) | dz | , ˜ λ ( z ) | dz | : = T ∗ λ ( z ) | dz | . Then ˜ µ ( z ) | dz | is a conformal pseudometric with curvature κ ( T ( z )) ∈ [ − c , − ] and ˜ λ ( z ) | dz | isa conformal pseudometric with curvature κ λ ( T ( z )) ≤ κ ˜ µ ( z ) such that˜ λ ( ) ˜ µ ( ) = λ ( z ) | T ′ ( z ) | µ ( z ) | T ′ ( z ) | < . By continuity, ˜ λ ( ξ ) / ˜ µ ( ξ ) < | ξ | = r provided that r ∈ ( , ) is sufficientlysmall. Hence, we can apply (5.1) for ˜ λ and ˜ µ and deduce˜ λ ( z ) / ˜ µ ( z ) < r ≤ | z | < r >
0. This proves λ / µ < D and (5.2) follows immedi-ately from (5.1). (cid:3) Proof of Corollary 5.4.
Choose ρ ∈ ( R , ) . Then Theorem 5.1 applied to the conformal pseu-dometrics ρλ ( ρ z ) | dz | and ρµ ( ρ z ) | dz | implies log λ ( ρ z ′ ) µ ( ρ z ′ ) ≤ C r / ρ (cid:18) − | z ′ | − ( r / ρ ) (cid:19) c ρ / max | ξ | = r / ρ log λ ( ρξ ) µ ( ρξ ) , r / ρ ≤ | z ′ | < , where c ρ = − min | z ′ |≤ κ µ ( ρ z ′ ) = − min | z |≤ ρ κ ( z ) . Replacing ρ z ′ by z so that r ≤ | z | ≤ R < ρ proves (5.4) with C = C r / ρ (cid:18) ρ − R ρ − r (cid:19) c ρ / = exp (cid:18) − ρ r (cid:19) (cid:18) ρ − R ρ − r (cid:19) c ρ / by (5.3). (cid:3) Proof of Theorem 3.2. If λ µ , then Theorem 5.1 implieslim sup n → ∞ log λ ( z n ) µ ( z n ) ( − | z n | ) c / < , which contradicts λ ( z n ) µ ( z n ) = + o (cid:16) ( − | z n | ) c / (cid:17) . (cid:3) IGIDITY 19
Remark . Theorem 5.1 extends a series of earlier results due to Golusin (see [22, Theorem3] or [23, p. 335], and Yamashita [51, 52], Beardon [6], Beardon & Minda [7]) and Chen [18].All these results are concerned with the special case µ = λ D and either κ λ ≡ − κ λ ≤ − λ D ( z ) | dz | . In particular, these resultsdo not cover for instance the case of the Nehari–Schwarz lemma, but Theorem 5.1 does. Theyare sufficient, however, for proving for instance the boundary Schwarz–Pick lemma (Theorem2.1) in the same way as Theorem 5.1 can be used to prove Theorem 3.2.In fact, the result of Golusin [22] is of a sligthly different nature when compared to (5.1). Itcan be rephrased as(5.9) λ ( z ) λ D ( z ) ≤ λ ( ) + | z | + | z | + λ ( ) | z | + | z | for all | z | < , for every conformal pseudometric λ ( z ) | dz | with curvature κ λ = −
4. This inequality has beenrediscovered many years later by Yamashita [51, 52] and independently by Beardon [6] andBeardon–Minda [7] as part of their elegant work on multi–point Schwarz–Pick lemmas. Withhindsight, Golusin’s inequality (5.9) is exactly the case w = λ / λ D on the entire unit disk in terms of its values at the originwhile (5.1) provides an estimate for λ / λ D “only” on each annulus r ≤ | z | < | z | = r . The reason for this difference is that (5.1) is validfor all pseudometrics λ ( z ) | dz | with curvature ≤ − λ ( z ) | dz | with constant curvature −
4. In fact, the following result shows thatfor the case κ λ ≤ − λ / λ D on the entire diskin terms of z and of λ ( ) which for λ ( ) < λ / λ D ≤ ≤ − Proposition 5.7.
For each a ∈ ( , ] let F a : = { λ ( z ) | dz | : λ ( z ) | dz | conformal pseudometric on D s.t. κ λ ≤ − and λ ( ) ≤ a } . Then sup λ ∈ F a λ ( z ) = λ D ( z ) for every z ∈ D \ { } .Proof. Let λ a ( z ) : = sup λ ∈ F a λ ( z ) , z ∈ D . Since obviously a λ D ( z ) | dz | belongs to F a we have λ a ( ) = a and λ a ( z ) ≥ a λ D ( z ) for all z ∈ D .Using the Perron machinery (see [28, Section 2.5]) it is not difficult to show that λ a ( z ) | dz | is aconformal metric with constant curvature − punctured disk D \ { } . A result of Nitsche[41] then implies that λ a ( z ) | dz | extends continuously to a conformal metric µ a ( z ) | dz | on theentire disk D with constant curvature − µ a ( z ) ≥ λ D ( z ) for all z ∈ D whereas Ahlfors’ lemma itself shows µ a ≤ λ D in D . In particular, λ a = µ a = λ D on D \ { } . (cid:3) Open problem . Perhaps there is a sharpening of the Nehari–Schwarz lemma in the spirit ofGolusin’s sharpening of the Schwarz–Pick lemma: Let B be a (finite or not) maximal Blaschkeproduct and f a holomorphic selfmap of D such that f ′ / B ′ has a holomorphic extension to D . Isthere an upper bound for f h ( z ) / B h ( z ) in terms of f h ( ) / B h ( ) and | z | which holds for all z ∈ D and which is better than f h / B h ≤ f h ( ) / B h ( ) < ROOF OF T HEOREM
URWITZ ’ S THEOREM FOR CONFORMAL PSEUDOMETRICS
The main additional technical tool for proving Theorem 4.4 is the following rigidity prop-erty of zeros of a sequence of pseudometrics. It is reminiscent of Hurwitz’s theorem aboutpreservation of zeros of holomorphic functions.
Theorem 6.1 (Rigidity of zeros) . Let λ n ( z ) | dz | , n = , , . . . , and µ ( z ) | dz | be conformal pseu-dometrics on D so that each λ n ( z ) | dz | has only isolated zeros and κ µ = κ for some locallyH¨older continuous function κ : D → R with κ ( z ) ≤ − in D . Suppose that λ n (cid:22) µ for any n ∈ N and lim n → ∞ λ n ( z n ) µ ( z n ) = for some sequence ( z n ) in D with z n → z ∈ D . Then for every ξ ∈ D the following hold:(a) If β ≥ resp. β n ≥ denotes the order of ξ as a zero of of µ ( z ) | dz | resp. λ n ( z ) | dz | , then β n → β . (b) If λ n ( z ) | dz | has a zero ξ n ∈ D \ { ξ } of order α n ≥ such that ξ n → ξ , then α n → . The proof of Theorem 6.1 is based on the Harnack inequality of Theorem 5.1 and the follow-ing auxiliary Lemma 6.3, which is a consequence of the Poisson–Jensen formula. Recall thatfor any R ∈ ( , ) Green’s function g R ( z , w ) for the disk K R ( ) = { z ∈ C : | z | < R } is given by g R ( z , w ) : = − log (cid:12)(cid:12)(cid:12)(cid:12) R ( z − w ) R − wz (cid:12)(cid:12)(cid:12)(cid:12) . We start with the following variant of the Poisson–Jensen formula.
Lemma 6.2 (Poisson–Jensen formula for conformal pseudometrics with isolated zeros) . Let λ ( z ) | dz | be a conformal pseudometric in D with κ λ ≤ − and only isolated zeros ξ , ξ , . . . ∈ D IGIDITY 21 with orders α > , α > , . . . . Then, for any R ∈ ( , ) , the subharmonic function log λ has aleast harmonic majorant h R on K R ( ) such thath R ( z ) ≤ log 11 − R , | z | < R , and log λ ( z ) = − ∑ | ξ j | < R α j g R ( z , ξ j ) + h R ( z ) + π ZZ K R ( ) g R ( z , w ) κ λ ( w ) λ ( w ) dA w , | z | < R . Recall our convention that κ λ is bounded (below) in some some disk K r ( ξ ) \ { ξ } for any ofits isolated zeros ξ . In particular, the area integral converges. Proof.
Since u : = log λ is C on D \ { ξ , ξ , . . . } and ∆ u = − κ λ ( z ) e u ≥ e u ≥ u issubharmonic on the entire disk D . By the Ahlfors–Schwarz lemma, λ ( z ) ≤ λ D ( z ) = − | z | , z ∈ D . Thus on the disk K R ( ) the function u ( z ) has the (constant) harmonic majorant − log ( − R ) ,and hence a least harmonic majorant h R : K R ( ) → R , say. We now consider v ( z ) : = u ( z ) + ∑ | ξ j | < R α j g R ( z , ξ j ) . By Theorem 1.1 in [29], which requires that κ λ is bounded on K R ( ) \ { ξ , ξ , . . . } , the subhar-monic function v is of class C on K R ( ) and ∆ v = − κ λ λ on K R ( ) \ { ξ , ξ , . . . } . It is easyto see that h R is the least harmonic majorant for v on K R ( ) , so the standard Poisson–Jensenformula for v (see, for instance, [29, Proposition 4.1]) proves the lemma. (cid:3) Lemma 6.3.
Let λ ( z ) | dz | and µ ( z ) | dz | be conformal pseudometrics on D with only isolatedzeros and κ µ = κ for some continuous function κ : D → R with κ ( z ) ≤ − in D . Suppose λ isdominated by µ . Then for any r ∈ ( , ) and ξ ∈ K r ( ) , log λ ( z ) µ ( z ) ≤ − ( α − β ) · g r ( z , ξ ) + r c r ( − r ) , | z | < r , where α ≥ resp. β ≥ denote the order of ξ as a zero of λ ( z ) | dz | resp. µ ( z ) | dz | , andc r : = − min | z |≤ r κ ( z ) . Proof.
Let ξ , ξ , . . . denote the pairwise distinct zeros of λ ( z ) | dz | with positive order α , α , . . . .Lemma 6.2 shows that for any r ∈ ( , ) ,log λ ( z ) = − ∑ | ξ j | < r α j g r ( z , ξ j ) + h λ , r ( z ) + π ZZ K r ( ) g r ( z , w ) κ λ ( w ) λ ( w ) dA w , | z | < r , where h λ , r denotes the least harmonic majorant of log λ on K r ( ) . In a similar way, we havelog µ ( z ) = − ∑ | ξ j | < r β j g r ( z , ξ j ) + h µ , r ( z ) + π ZZ K r ( ) g r ( z , w ) κ µ ( w ) µ ( w ) dA w , | z | < r , where β j ≥ ξ j as a zero of λ ( z ) | dz | and h µ , r denotes the least harmonic majorantof log µ on K r ( ) . From λ (cid:22) µ we get h λ , r ≤ h µ , r and β j ≤ α j for j = , , . . . . Therefore, if | ξ j | < r , thenlog λ ( z ) µ ( z ) ≤ − (cid:0) α j − β j (cid:1) g r ( z , ξ j ) + π ZZ K r ( ) g r ( z , w ) (cid:2) κ λ ( w ) λ n ( w ) − κ µ ( w ) µ ( w ) (cid:3) dA w ≤ − ( α j − β j ) g r ( z , ξ j ) − π ZZ K r ( ) g r ( z , w ) κ µ ( w ) µ ( w ) dA w ≤ − ( α j − β j ) g r ( z , ξ j ) + π ZZ K r ( ) g r ( z , w ) dA w · c r ( − r ) , where c r : = − min | z |≤ r κ ( z ) . In the last step we have used Ahlfors’ lemma which gives µ ( w ) ≤ λ D ( w ) = − | w | ≤ − r for all | w | ≤ r . Since 12 π ZZ K r ( ) g r ( z , w ) dA w = r − | z | , the proof is complete. (cid:3) Proof of Theorem 6.1.
Replacing λ n by T ∗ λ n and µ by T ∗ µ where T ( z ) = z − z − z z , we may assume from now on that z n → z = λ n (cid:22) µ , we have β n ≥ β . We first consider the case ξ = λ n ( z ) ≤ ( + β n ) | z | β n − | z | ( + β n ) , | z | < . This follows from the easily verified fact that ( + β n ) | z | β n − | z | ( + β n ) | dz | IGIDITY 23 is the maximal conformal pseudometric on D with curvature = − β n at z =
0. Hence we get λ n ( z n ) µ ( z n ) ≤ | z n | β n − β | z n | β µ ( z n ) + β n − | z n | ( + β n ) . Therefore, in view of β n ≥ β and Lemma 3.8, the condition λ n ( z n ) / µ ( z n ) → β n → β ,and (a) is proved. In order to prove (b) we fix r ∈ ( , ) and use Lemma 6.3. Note that theassumption λ n (cid:22) µ implies that either µ ( ξ ) = ξ is an isolated zero of µ ( z ) | dz | , so µ ( ξ n ) = n . We therefore get from Lemma 6.3,log λ n ( z n ) µ ( z n ) ≤ − α n g r ( z n , ξ n ) + r c r ( − r ) = α n log (cid:12)(cid:12)(cid:12)(cid:12) r ( z n − ξ n ) r − z n ξ n (cid:12)(cid:12)(cid:12)(cid:12) + r c r ( − r ) . Since λ n ( z n ) / µ ( z n ) → ξ n → z n →
0, we deduce α n → ξ = T n ( z ) : = z n − ξ − z − ξ z − z n ξ − z − ξ z , for which T n ( ξ ) = z n → T n ( z n ) → − ξ . Then ˜ λ n : = ( T − n ) ∗ λ n (cid:22) µ n : = ( T − n ) ∗ µ . Finallywe fix two positive constants R < ρ < < | ξ | < R < ρ < γ : = sup n ∈ N (cid:18) − min | z |≤ ρ κ µ n ( z ) (cid:19) = sup n ∈ N (cid:18) − min | z |≤ ρ κ µ ( T − n ( z )) (cid:19) < ∞ . In order to prove β n → β we argue by contradiction and, recalling β n ≥ β , we assume β ′ : = lim sup n → ∞ β n > β . First we choose ˜ r > r < | ξ | and(6.1) (cid:18) (cid:19) β ′ − β exp (cid:18) ˜ r γ ( − ˜ r ) (cid:19) < . Next we choose a positive integer N such that | T n ( ξ ) | < r
11 and ˜ r ≤ | T n ( z n ) | < R for all n ≥ N . This is possible since T n ( ξ ) → T n ( z n ) → − ξ , where 0 < | ξ | < R . It is easy to see that thischoice of N ensures in particular exp ( − g ˜ r ( z , T n ( ξ ))) ≤ for all | z | ≤ ˜ r / n ≥ N . Since T n ( ξ ) ∈ K ˜ r ( ) is a zero of ˜ λ n ( z ) | dz | of order β n and a zeroof µ n ( z ) | dz | of order β , Lemma 6.3 applied to the disk K ˜ r ( ) therefore implies˜ λ n ( z ) µ n ( z ) ≤ (cid:18) (cid:19) β n − β exp (cid:18) ˜ r γ ( − ˜ r ) (cid:19) , | z | ≤ ˜ r / , n ≥ N . In view of r : = ˜ r / ≤ | T n ( z n ) | < R for all n ≥ N , Corollary 5.4 now gives us(6.2) log λ n ( z n ) µ ( z n ) = log ˜ λ n ( T n ( z n )) µ n ( T n ( z n )) ≤ C n log "(cid:18) (cid:19) β n − β exp (cid:18) ˜ r γ ( − ˜ r ) (cid:19) with C n = C ( r , R , κ µ n ) bounded from below by the positive numberexp (cid:18) − ρ r (cid:19) · (cid:18) ρ − R ρ − r (cid:19) γ / , see Remark 5.5. In view of (6.1), we thus see that inequality (6.2) contradicts λ n ( z n ) / µ ( z n ) → β n → β .In order to prove (b), we proceed in a similar way, but now we consider the unit disk auto-morphisms S n ( z ) : = z n − ξ n − z − ξ n z − z n ξ n − z − ξ n z , for which S n ( ξ n ) = z n → S n ( z n ) → − ξ . As above we have ˜ λ n : = ( S − n ) ∗ λ n (cid:22) µ n : =( S − n ) ∗ µ . We fix constants R > ρ > < | ξ | < R < ρ <
1. Then, as above, γ ′ : = sup n ∈ N (cid:18) − min | z |≤ ρ κ µ n ( z ) (cid:19) = sup n ∈ N (cid:18) − min | z |≤ ρ κ µ ( S − n ( z )) (cid:19) < ∞ . In order to prove α n → α ′ : = lim sup n → ∞ α n > , we can choose 0 < ˜ r < | ξ | such that(6.3) (cid:18) (cid:19) α ′ exp (cid:18) ˜ r γ ( − ˜ r ) (cid:19) < . Having fixed ˜ r > N with | S n ( ξ n ) | < r
11 and ˜ r ≤ | S n ( z n ) | < R for all n ≥ N , IGIDITY 25 as well as(6.4) µ ( ξ n ) = n ≥ N . This is possible since S n ( ξ n ) = z n → S n ( z n ) → − ξ with ˜ r < | ξ | < R , and since either µ ( ξ ) = ξ is an isolated zero of µ ( z ) | dz | by our assumption λ n (cid:22) µ . This choice of N implies exp ( − g ˜ r ( z , z n )) ≤
34 for all | z | ≤ ˜ r n ≥ N . Since z n = S n ( ξ n ) ∈ K ˜ r ( ) is a zero of order α n ≥ λ n ( z ) | dz | , but not a zero of µ n ( z ) | dz | by(6.4), Lemma 6.3 applied for K ˜ r ( ) implies˜ λ n ( z ) µ n ( z ) ≤ (cid:18) (cid:19) α n exp (cid:18) ˜ r γ ( − ˜ r ) (cid:19) , | z | ≤ ˜ r / , n ≥ N . Since r : = ˜ r / < | S n ( z n ) | < R , Corollary 5.4 showslog λ n ( z n ) µ ( z n ) = log ˜ λ n ( S n ( z n )) µ n ( S n ( z n )) ≤ C n log (cid:20)(cid:18) (cid:19) α n exp (cid:18) γ ( − ˜ r ) ˜ r (cid:19)(cid:21) , n ≥ N , with C n bounded below by some positive constant independent of n , as above. In view of (6.3)this inequality contradicts λ n ( z n ) / µ ( z n ) →
1, and we have shown α n → (cid:3) Proof of Theorem 4.4 for the interior case: ( z n ) is compactly contained in D . We denote G : = { z ∈ D : µ ( z ) > } , so D \ G is the set of zeros of µ ( z ) | dz | , which are isolated by assumption.We distinguish two cases.1. Case: There is a subsequence ( n k ) such that λ n k ( z ) | dz | has a zero ξ n k ∈ G of order α n k > ξ n k → ξ ∈ D . Passing to another subsequence, if necessary, we may assume z n k → z ∈ D . If ξ ∈ G , then ξ is not a zero of µ ( z ) | dz | , so α n k → ξ G , then ξ n k = ξ for all but finitely many k , so α n k → R ∈ ( , ) there is a positive integer N = N ( R ) such that for any n ≥ N thepseudometric λ n ( z ) | dz | has no zeros in G R : = K R ( ) ∩ G . We show that (i) holds.We first prove that { λ n / µ : n ∈ N } is a normal family of continuous functions on each disk K R ( ) . Fix R ∈ ( , ) , let n ≥ N ( R ) and let ξ , . . . , ξ L be the zeros of µ ( z ) | dz | in K R ( ) of order β , . . . , β L >
0, say. Then the points ξ , . . . , ξ L are exactly the zeros of λ n ( z ) | dz | in K R ( ) oforder β , n ≥ β , . . . β L , n ≥ β L . Let h n , R and h R denote the least harmonic majorant of log λ n , R and log µ in K R ( ) . Lemma 6.2 showslog λ n ( z ) µ ( z ) = − L ∑ j = (cid:0) β j , n − β j (cid:1) g R ( z , ξ j ) + h n , R ( z ) − h R ( z )+ π ZZ K R ( ) g R ( z , w ) (cid:2) κ λ n ( w ) λ n ( w ) − κ µ ( w ) µ ( w ) (cid:3) dA w = : − L ∑ j = (cid:0) β j , n − β j (cid:1) g R ( z , ξ j ) + h n , R ( z ) − h R ( z ) + p n ( z ) , | z | < R , where h n , R belongs to the set of all harmonic functions in K R ( ) which are bounded bylog ( / ( − R ) , so { h n , R : n ≥ N ( R ) } is a normal family on K R ( ) (see [45, Theorem 5.4.2]).Again using the Ahlfors–Schwarz lemma, we see that the Green potential p n ( z ) satisfies | p n ( z ) − p n ( z ) | ≤ π ZZ K R ( ) | g R ( z , w ) − g R ( z , w ) | dA · c R ( − R ) as well as | p n ( z ) | ≤ π ZZ K R ( ) g R ( z , w ) dA · c R ( − R ) for all z , z , z ∈ K R ( ) . Here c R is a positive constant independent of n such that | κ µ ( z ) | ≤ c R and | κ λ n ( z ) | ≤ c R for all n = , . . . and all | z | ≤ R . Using the explicit expression for theGreen’s function g R , we deduce that the family { p n : n ≥ N ( R ) } of all such Green potentialsis locally uniformly equicontinuous and locally uniformly bounded on K R ( ) . Finally, there isa subsequence ( z n k ) which tends to some z ∈ D , so β j , n k → β j as k → ∞ for each j = , . . . , L by Theorem 6.1 (a). We conclude that { λ n / µ : n ≥ N ( R ) } is a normal family in K R ( ) for each R ∈ ( , ) and hence on D .We now prove that λ n / µ → D . There is a subsequence of ( λ n k / µ ) which converges locally uniformly in D to some continuous function g : D → R with 0 ≤ g ( z ) ≤ z ∈ D . If z ∈ D denotes a subsequential limit of ( z n k ) , then (4.1) implies g ( z ) = g ( z ) < z ∈ D . Consider˜ g : = g ◦ T with T ( z ) : = z + z + z z . Then ˜ g ( ) < g is continuous there is r ∈ ( , ) such that ˜ g ( z ) < | z | ≤ r . Inparticular, R : = | T − ( z ) | > r . We now use Corollary 5.4, which shows that T ∗ λ n ( z ) T ∗ µ ( z ) ≤ max | ξ | = r (cid:18) T ∗ λ n ( ξ ) T ∗ µ ( ξ ) (cid:19) C on r ≤ | z | ≤ R . IGIDITY 27
The Harnack constant C > n . Hence we get˜ g ( z ) ≤ max | ξ | = r ˜ g ( ξ ) C < r ≤ | z | ≤ R . Using this inequality for the point T − ( z ) we have 1 = g ( z ) = ˜ g ( T − ( z )) <
1, a contradiction.We have shown that every subsequential locally uniform limit function of ( λ n / µ ) is the constantfunction 1. Since ( λ n / µ ) is a normal family on D , we see that λ n / µ → D . (cid:3) Proof of Theorem 4.4 for the boundary case: | z n | → . By assumption,lim n → ∞ log λ n ( z n ) µ ( z n ) ( − | z n | ) c / = . Hence Theorem 5.1 implies that for any r ∈ ( , ) ,lim n → ∞ max | ξ | = r log λ n ( ζ ) µ ( ζ ) = . Thus for fixed r = /
2, say, and any n = , , . . . there is a point z ′ n such that | z ′ n | = r andlim n → ∞ λ n ( z ′ n ) µ ( z ′ n ) = . Hence the boundary case of Theorem 4.4 follows from the interior case of Theorem 4.4, whichwe have already established. (cid:3)
It is clear that the two alternatives (i) and (ii) of Theorem 4.4 are mutually exclusive. In fact,if both conditions (i) and (ii) hold, then this would imply µ ( ξ ) =
0. Since by assumption, µ ( z ) | dz | has only isolated zeros, µ ( z ) > z ∈ K r ( ξ ) \ { ξ } for some r >
0. But then (i)forces that λ n / µ never vanishes in K r ( ξ ) \ { } for all but finitely many n , contradicting (ii).7. I NFINITESIMAL RIGIDITY IN STRONGLY CONVEX DOMAINS
Complex geodesics in strongly convex domains.
In this section we recall some resultson complex geodesics in bounded strongly convex domains with smooth boundary as neededfor our aims.In all this section, Ω ⊂ C N is a bounded strongly convex domain with smooth boundary.For ζ ∈ ∂Ω , let us denote by T C ζ ∂Ω the complex tangent space of ∂Ω at ζ . In other words, T C ζ ∂Ω : = T ζ ∂Ω ∩ i ( T ζ ∂Ω ) . We denote by K Ω ( z , w ) the Kobayashi distance between z , w ∈ Ω . We refer the reader to thebooks [1, 27] for some definitions and details.We recall the classical boundary estimates of the Kobayashi distance in strongly (pseudo)-convex domains (see, e.g., [1, Thm. 2.3.51, Thm. 2.3.52]). Proposition 7.1.
Let Ω ⊂ C N be a bounded strongly convex domain with smooth boundary andlet p ∈ Ω . Then there exists C > such that for all z ∈ Ω −
12 log δ Ω ( z ) − C ≤ K Ω ( p , z ) ≤ −
12 log δ Ω ( z ) + C A complex geodesic is a holomorphic map ϕ : D → Ω which is an isometry between thehyperbolic distance K D of D = { ζ ∈ C | | ζ | < } and the Kobayashi distance K D in D . Aholomorphic map h : D → Ω is a complex geodesic if and only if it is an infinitesimal isometryat one—and hence any—point between the Poincar´e metric k D of D and the Kobayashi metric k Ω of Ω (see [1, Ch. 2.6]).According to Lempert (see [32, 33, 34] and [1]), any complex geodesic extends smoothly tothe boundary of the disc and ϕ ( ∂ D ) ⊂ ∂Ω . Moreover, given any two points z , w ∈ Ω , z = w ,there exists a complex geodesic ϕ : D → Ω such that z , w ∈ ϕ ( D ) . Such a geodesic is unique upto pre-composition with automorphisms of D . Conversely, if ϕ : D → Ω is a holomorphic mapsuch that K Ω ( ϕ ( ζ ) , ϕ ( ζ )) = K D ( ζ , ζ ) for some ζ = ζ ∈ D , then ϕ is a complex geodesic.If ϕ : D → Ω is a complex geodesic, then for every θ ∈ R , ϕ ′ ( e i θ ) ∈ C N \ T C ϕ ( e i θ ) ∂Ω . Con-versely, if z ∈ Ω and v ∈ C N \ { O } (and v T C z ∂Ω if z ∈ ∂Ω ) there exists a unique (still, up topre-composition with automorphisms of D ) complex geodesic ϕ : D → Ω such that z ∈ ϕ ( D ) and ϕ ( D ) is parallel to v (in case z , w ∈ ∂Ω this follows from Abate [2] and Chang, Hu andLee [16]).If ϕ : D → Ω is a complex geodesic then there exists a (unique when suitably normalized)holomorphic map ˜ ρ : Ω → D , smooth up to ∂Ω such that ˜ ρ ◦ ϕ = id D . The map ˜ ρ is called the left inverse of ϕ . It is known that ˜ ρ − ( e i θ ) = { ϕ ( e i θ ) } for all θ ∈ R , while the fibers e ρ − ( ζ ) are the intersection of Ω with affine complex hyperplanes for all ζ ∈ D (see, e.g., [12, Section3]). The map ρ : = ϕ ◦ ˜ ρ : D → ϕ ( D ) is called the Lempertprojection.In the sequel we shall use the following results which follow from, e.g., [12, Corollary 2.3,Lemma 3.5]: Proposition 7.2.
Let Ω ⊂ C N be a bounded strongly convex domain with smooth bound-ary. Let { ϕ k } k ∈ N be a family of complex geodesics of Ω parameterized so that δ Ω ( ϕ k ( )) = max ζ ∈ D δ Ω ( ϕ k ( ζ )) .(1) If there is c > such that δ Ω ( ϕ k ( )) ≥ c for all k, then, up to extracting a subsequence, ϕ k converges uniformly on D together with all its derivatives to a complex geodesic ϕ : D → Ω .(2) If there is a sequence t k ∈ ( , ) converging to such that lim k → ∞ ϕ k ( t k ) = ξ ∈ ∂Ω and lim k → ∞ ϕ ′ k ( t k ) | ϕ ′ k ( t k ) | = v ∈ C N \ T C ξ ∂Ω , then there is c > such that δ Ω ( ϕ k ( )) ≥ c for all k.(3) If ϕ k converges uniformly on D to a complex geodesic ϕ : D → Ω , then the sequence { ρ k } of Lempert projections converges uniformly on Ω , together with all its derivatives, to theLempert projection of ϕ . IGIDITY 29
Rigidity results. If ζ ∈ ∂Ω and v ∈ C N , we let Π ζ ( v ) be the orthogonal projection of v onto T C ζ ∂Ω .Moreover, for z ∈ Ω , we also let π ( z ) ∈ ∂Ω be such that | z − π ( z ) | = δ Ω ( z ) . The choiceof π ( z ) might not be unique in general, but, if z is sufficiently close to ∂Ω , then π ( z ) is welldefined. Lemma 7.3.
Let D , D ′ ⊂ C N be two bounded strongly convex domains with smooth boundary.Let F : D → D ′ be holomorphic. Let p ∈ ∂ D. Let ϕ : D → D be a complex geodesic such that ϕ ( ) = p. Suppose that there exists an increasing sequence { r n } ⊂ ( , ) converging to suchthat,— if { F ( ϕ ( r n k )) } is converging to some q ∈ ∂ D ′ then (7.1) lim sup k → ∞ | Π π ( F ( ϕ ( r nk ))) ( dF ϕ ( r nk ) ( ϕ ′ ( r n k ))) | < + ∞ , — and (7.2) k D ′ ( F ( ϕ ( r n )) ; dF ϕ ( r n ) ( ϕ ′ ( r n ))) = k D ( ϕ ( r n ) ; ϕ ′ ( r n )) + o ( δ D ( ϕ ( r n ))) . Then D ∋ ζ F ( ϕ ( ζ )) ∈ D ′ is a complex geodesic in D ′ and F | ϕ ( D ) : ϕ ( D ) → F ( ϕ ( D )) is abiholomorphism.Proof. Let us denote v n : = ϕ ′ ( r n ) and x n : = | dF ϕ ( r n ) ( ϕ ′ ( r n )) | . Note that, by (7.2), x n > w n : = dF ϕ ( rn ) ( v n ) x n , δ n : = δ D ( ϕ ( r n )) and δ ′ n : = δ D ′ ( F ( ϕ ( r n ))) .We consider two cases, which we can always reduce to: either there exists C > x n ≤ C for all n , or lim n → ∞ x n = ∞ .Case 1. There exists C > x n ≤ C for all n . We know that v n → v for some v ∈ C N \ T C p ∂ D , and k D ( r n ; 1 ) = k D ( ϕ ( r n ) ; ϕ ′ ( r n )) . Hence, k D ( ϕ ( r n ) ; ϕ ′ ( r n )) → ∞ as n → ∞ , and therefore by (7.2), k D ′ ( F ( ϕ ( r n )) ; dF ϕ ( r n ) ( ϕ ′ ( r n ))) = x n k D ′ ( F ( ϕ ( r n )) ; w n ) converges to ∞ as well. Since | w n | ≡ x n ≤ C , it follows that { F ( ϕ ( r n )) } is not relativelycompact in D ′ , and, up to subsequences, we can assume it converges to some q ∈ ∂ D ′ .If z ∈ D ′ is sufficiently close to ∂ D ′ there exists a unique π ( z ) ∈ ∂ D ′ closest to z , and for w ∈ T z D ′ = C N , there is a unique orthogonal decomposition w = w T + Π π ( z ) ( w ) . In order toavoid burdening notation, we set w ⊥ : = Π π ( z ) ( w ) in the rest of the proof.By Aladro’s estimates (see [4], see also [24, 37]), we have(7.3) k D ′ ( F ( ϕ ( r n )) ; w n ) ∼ (cid:18) | w ⊥ n | δ ′ n + | w Tn | ( δ ′ n ) (cid:19) / , where, as customary, we use here the following notation: if f ( n ) and g ( n ) are functions depend-ing on n , we write f ( n ) ∼ g ( n ) provided there exists C > C ≤ f ( n ) g ( n ) ≤ C for all n . By the same token, we have(7.4) k D ( ϕ ( r n ) ; v n ) ∼ (cid:18) | v ⊥ n | δ n + | v Tn | ( δ n ) (cid:19) / ∼ δ n , where for the last ∼ we used the fact that v n → v for some v ∈ C n \ T C p ∂ D , hence v Tn → v T = k D ′ ( F ( ϕ ( r n )) ; x n w n ) ∼ k D ( ϕ ( r n ) ; v n ) , hence, by (7.3) and (7.4),(7.5) x n (cid:18) | w ⊥ n | δ ′ n + | w Tn | (cid:19) / ∼ δ ′ n δ n . Now, let p ∈ D and q : = F ( p ) ∈ D ′ . By Proposition 7.1, there exists C ∈ R such that for all n , 12 log δ ′ n δ n ≥ K D ( p , ϕ ( r n )) − K D ′ ( q , F ( ϕ ( r n ))) + C ≥ C > − ∞ . Hence, there exists c > δ ′ n δ n ≥ c for all n . Therefore, from (7.5) (since x n is boundedand | w ⊥ n | , | w n | T ≤ n ), we obtain thatlim inf n → ∞ x n > , lim inf n → ∞ | w Tn | > . In particular, up to subsequences, we can assume that w n → w ∈ C N , and, since w Tn → w T = w T C q ∂ D ′ .For every n , let η n : D → D ′ be a complex geodesic such that δ D ′ ( η n ( )) = max ζ ∈ D δ D ′ ( η n ( ζ )) ,that F ( ϕ ( r n )) = η n ( t n ) for some t n ∈ ( , ) and η ′ n ( t n ) = λ n w n for some λ n >
0. Let ρ n : D ′ → η n ( D ) be the Lempert projection associated with η n .Since w n → w ∈ C N \ T C q ∂ D ′ , by Proposition 7.2, up to extracting subsequences, { η n } con-verges uniformly on D to a complex geodesic η : D → D ′ . Since η n ( t n ) → q , it follows that η ( ) = q . Moreover, η − n ◦ ρ n converges uniformly on D to η − ◦ ρ , where ρ is the Lempertprojection associated with η . Therefore, if we let f n : = η − n ◦ ρ n ◦ F ◦ ϕ : D → D , we have that f n converges uniformly on compacta of D to the holomorphic function f : = η − ◦ ρ ◦ F ◦ ϕ : D → D . Moreover, taking into account that, by construction, ρ n ( F ( ϕ ( r n ))) = F ( ϕ ( r n )) and d ( ρ n ) F ( ϕ ( r n )) ( dF ϕ ( r n ) ( ϕ ′ ( r n ))) = dF ϕ ( r n ) ( ϕ ′ ( r n )) , IGIDITY 31 since η n are complex geodesics, we have by (7.2), f hn ( r n ) = k D ( f n ( r n ) ; f ′ n ( r n )) k D ( r n ; 1 ) = k D ′ ( ρ n ( F ( ϕ ( r n ))) ; d ( ρ n ) F ( ϕ ( r n )) ( dF ϕ ( r n ) ( ϕ ′ ( r n )))) k D ( ϕ ( r n ) ; ϕ ′ ( r n ))= k D ′ ( F ( ϕ ( r n )) ; dF ϕ ( r n ) ( ϕ ′ ( r n ))) k D ( ϕ ( r n ) ; ϕ ′ ( r n )) = + o ( δ n ) k D ( ϕ ( r n ) ; ϕ ′ ( r n )) . (7.6)By Proposition 7.1, there exist C , C ∈ R such that for all n , C = [ K D ( , r n ) − K D ( ϕ ( ) , ϕ ( r n ))] − C ≤
12 log δ n − r n ≤ [ K D ( , r n ) − K D ( ϕ ( ) , ϕ ( r n ))] + C = C . Therefore, δ n ∼ ( − r n ) . Moreover, k D ( ϕ ( r n ) ; ϕ ′ ( r n )) = k D ( r n ; 1 ) = − r n . Thus, by (7.6), we have f hn ( r n ) = + o (( − r n ) ) . Since we already know that the limit f of { f n } is not an unimodular constant, it follows fromCorollary 4.6 that f is an automorphism of D .Since f = η − ◦ ρ ◦ F ◦ ϕ , we have η ◦ f = ρ ◦ F ◦ ϕ . Taking into account that η ◦ f is anisometry between K D and K D ′ , we have for all ζ , ζ ′ ∈ D , K D ( ζ , ζ ′ ) ≥ K D ′ ( F ( ϕ ( ζ )) , F ( ϕ ( ζ ′ ))) ≥ K D ′ ( ρ ( F ( ϕ ( ζ ))) , ρ ( F ( ϕ ( ζ ′ ))))= K D ′ ( η ( f ( ζ )) , η ( f ( ζ ′ ))) = K D ( ζ , ζ ′ ) . Hence, K D ′ ( F ( ϕ ( ζ )) , F ( ϕ ( ζ ′ ))) = K D ( ζ , ζ ′ ) for all ζ , ζ ′ ∈ D , and F ◦ ϕ : D → D ′ is a complexgeodesic and, clearly, F | ϕ ( D ) : ϕ ( D ) → F ( ϕ ( D )) is a biholomorphism.Case 2. lim n → ∞ x n = ∞ . We retain the notations introduced in Case 1.If { F ( ϕ ( r n )) } is relatively compact in D ′ (a case that, a posteriori, cannot occur), we canassume that { F ( ϕ ( r n )) } converges to some q ∈ D ′ and w n → w ∈ C N , | w | =
1. Therefore, { η n } converges uniformly on D to a complex geodesic η : D → D ′ such that η ( t ) = q , (where t = lim n → ∞ t n ∈ ( , ) ) and η ′ ( t ) = λ w for some λ >
0. Hence, arguing as in Case 1, we seethat { f n } converges uniformly on compacta to an automorphism f of D , F ◦ ϕ : D → D ′ is acomplex geodesic and F | ϕ ( D ) : ϕ ( D ) → F ( ϕ ( D )) is a biholomorphism.In case { F ( ϕ ( r n )) } is not relatively compact in D ′ , up to passing to a subsequence, wecan assume that { F ( ϕ ( r n )) } converges to some q ∈ ∂ D ′ and (7.1) holds for all r n ’s. Hence,lim n → ∞ x n = ∞ and lim sup n → ∞ | Π F ( ϕ ( r n )) ( dF ϕ ( r n ) ( v n )) | < + ∞ . Therefore,lim n → ∞ | w ⊥ n | = lim n → ∞ | Π F ( ϕ ( r n )) ( dF ϕ ( r n ) ( v n )) | x n = . It follows that w n → w ∈ C N \ T C q ∂ D ′ , and we can repeat the argument in Case 1 to end theproof. (cid:3) The same argument in the proof of Lemma 7.3 shows the following boundary infinitesimalcharacterization of complex geodesics:
Proposition 7.4.
Let Ω ⊂ C N be a bounded strongly convex domain with smooth boundary.Let f : D → Ω be holomorphic. Then f is a complex geodesic if and only if there exists anincreasing sequence { r n } ⊂ ( , ) converging to such that,— if { f ( r n k ) } is converging to some q ∈ ∂Ω then lim sup k → ∞ | Π π ( f ( r nk )) ( f ′ ( r n k )) | < + ∞ , — and k Ω ( f ( r n ) ; f ′ ( r n )) = − r n + o ( − r n ) . Now, we use the previous lemma to prove an infinitesimal boundary rigidity principle.Let D ⊂ C N be a bounded domain with smooth boundary. Let p ∈ ∂ D and v ∈ C N \ T C p ∂ D , | v | = D (namely, p + rv ∈ D for small r > α ∈ ( , ) , thecone C D ( p , v , α ) in D of vertex p , direction v and aperture α is C D ( p , v , α ) : = { z ∈ D : Re h z − p , v i > α | z − p |} , where h z , w i = ∑ Nj = z j w j is the standard Hermitian product in C N . Theorem 7.5.
Let D , D ′ ⊂ C N be two bounded strongly convex domains with smooth boundary.Let F : D → D ′ be holomorphic. Let p ∈ ∂ D. Then F is a biholomorphism if and only if(1) the cluster set of { F ( z n ) } belongs to ∂ D ′ for every { z n } ⊂ D converging to p and suchthat { ( z n − p ) / | z n − p |} converges to some τ ∈ T C p ∂ D;(2) for every v ∈ C N \ T C p ∂ D, | v | = pointing inside D there exist α ∈ ( , ) and C > suchthat for all w ∈ C N with | w | = and Re h w , v i > α ,a) if { z k } ⊂ C D ( p , v , α ) is a sequence converging to p such that { F ( z k ) } has no accumu-lation points in D ′ then lim sup k → ∞ | Π π ( F ( z k )) ( dF z k ( w )) | ≤ C , b) and k D ′ ( F ( z ) ; dF z ( w )) = k D ( z ; w ) + o ( δ D ( z )) , when C D ( p , v , α ) ∋ z → p, uniformly in w.Proof. Let ϕ : D → D be a complex geodesic such that ϕ ( ) = p . By Hopf’s Lemma, ϕ ′ ( ) ∈ C N \ T C p ∂ D . Moreover, { ϕ ( − / n ) } converges to p non-tangentially since, as in the proof ofthe previous lemma, Proposition 7.1 implies that δ D ( ϕ ( − / n )) ∼ δ D ( − / n ) = | − ( − / n ) | ∼ | p − ϕ ( − / n ) | , IGIDITY 33 where the last ∼ follows from ϕ extending smoothly on D . Therefore, hypothesis (2) impliesthat for n sufficiently large, the sequence { r n : = − / n } satisfies the hypotheses of Lemma 7.3.Hence, F | ϕ ( D ) : ϕ ( D ) → F ( ϕ ( D )) is a biholomorphism. By the arbitrariness of ϕ , it followsthat F maps complex geodesics of D containing p in the closure onto complex geodesics of D ′ acting as an isometry on them.Now we prove that F is proper. Suppose by contradiction that there exists a sequence { z k } ⊂ D converging to some ξ ∈ ∂ D and such that { F ( z k ) } is relatively compact in D ′ .Let ϕ k : D → D be a complex geodesic such that z k , p ∈ ϕ k ( D ) for all k . We can assume that ϕ k is parameterized so that δ D ( ϕ k ( )) = max ζ ∈ D δ D ( ϕ k ( ζ )) and ϕ k ( t k ) = z k for some t k ∈ ( , ) .We claim that there exists c > δ D ( ϕ k ( )) ≥ c for all k .Indeed, suppose this is not the case and, up to passing to a subsequence if necessary, assumethat δ D ( ϕ k ( )) → k → ∞ . Since ϕ k is smooth on D , it follows that ( z k − p ) / | z k − p | ∼ ϕ ′ k ( t k ) / | ϕ ′ k ( t k ) | . Hence, by Proposition 7.2, it turns out that the cluster set of { ( z k − p ) / | z k − p |} belongs to T C p ∂ D , but, by hypothesis (1), { F ( z k ) } cannot have accumulation points in D ′ , acontradiction.Hence, δ D ( ϕ k ( )) ≥ c for all k and { F ( ϕ k ( )) } is relatively compact in D ′ .For what we already proved, F ◦ ϕ k : D → D ′ is a complex geodesic in D ′ . Since { F ( ϕ k ( t k )) } and { F ( ϕ k ( )) } are relatively compact in D ′ , we obtain K D ( ϕ k ( t k ) , ϕ k ( )) = K D ( t k , ) = K D ′ ( F ( ϕ k ( t k )) , F ( ϕ k ( ))) < + ∞ . However, lim k → ∞ K D ( ϕ k ( t k ) , ϕ k ( )) = ∞ , since { ϕ k ( t k ) } converges to the boundary of D and { ϕ k ( ) } is relatively compact in D , obtaining a contradiction.Therefore, F is proper. By [19, Theorem 1] it follows that F is a biholomorphism. (cid:3) We can now prove Theorem 2.3 and Theorem 2.4.
Proof of Theorem 2.3.
The result is essentially a consequence of Theorem 7.5 and its proof.Indeed, the only issue is to see that for the ball one can reduce hypothesis (2) by allowing only z ∈ ( C v + e ) ∩ B N converging to e non-tangentially.However, we observe that for all v ∈ C N \ T C e ∂ B N , the set ( C v + e ) ∩ B N is (the imageof) a complex geodesic in B N (seee, e.g., [1]). Therefore, in order to apply Lemma 7.3, wejust need the limit in the second condition of (2) for z ∈ ( C v + e ) ∩ B N converging to e non-tangentially. (cid:3) Proof of Theorem 2.4.
Arguing as in the proof of Theorem 7.5, we can prove that for every { z k } ⊂ D such that lim k → ∞ z k = ξ ∈ ∂ D \ { p } then { F ( z k ) } has no accumulation points in D ′ .Assume that lim k → ∞ z k = p . Suppose by contradiction that { F ( z k ) } is relatively compact in D ′ . Let ϕ k : D → D be complex geodesics such that ϕ k ( ) = z k . Since F ◦ ϕ k : D → D ′ arecomplex geodesics for all k , we have k D ( ϕ k ( ) ; ϕ ′ k ( ) | ϕ ′ k ( ) | ) = k D ′ ( F ( ϕ k ( )) ; dF ϕ k ( ) ( ϕ ′ k ( ) | ϕ ′ k ( ) | )) . By hypothesis (1), | dF ϕ k ( ) ( ϕ ′ k ( ) | ϕ ′ k ( ) | ) | ≤ C for k sufficiently large, and { F ( ϕ k ( )) } is relativelycompact in D ′ , thus the right hand side of the previous equation is bounded. On the other hand,since { ϕ k ( ) } converges to p , the left hand side explodes, giving a contradiction. This provesthat F is proper and hence it is a biholomorphism. (cid:3) Remark . It is very likely that one can relax the technical hypotheses of Theorem 7.5, namely,one can probably remove hypothesis (1) and hypothesis (2).a), although, at present, we needthem in our proof.On the other hand, it seems however more complicated to extend the result to holomor-phic maps between strongly pseudoconvex domains, since our method is based on complexgeodesics and Lempert’s theory: the original Burns-Krantz result for strongly (pseudo)convexdomains was also based on complex geodesics, but, in that case, there was the advantage thatin order to prove that a map is the identity is enough to prove it is the identity on an open set,and it is known that complex geodesics at a given boundary point of a strongly pseudoconvexdomain exist and have holomorphic retractions for an open set of directions.8. A
PPENDIX : T
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Proposition 8.1.
Let f : D → D be a holomorphic function such thatf ( z ) = z + o (cid:0) | − z | (cid:1) as z → nontangentially . Then f h ( z ) = + o (cid:0) | − z | (cid:1) as z → nontangentially . Proof.
Let S be a sector in D with vertex at + α and S ′ a slightly largersector with vertex at + β . If we denote for z ∈ S by C ( z ) the circle withcenter z and radius r ( z ) = dist ( z , ∂ S ′ ) , then f ′ ( z ) = π i Z C ( z ) w − + ( f ( w ) − w )( w − z ) dw = π i Z C ( z ) dww − z + π i Z C ( z ) z − + f ( w ) − w ( w − z ) dw = + π i Z C ( z ) f ( w ) − w ( w − z ) dw = : 1 + I ( z ) . IGIDITY 35
Now for fixed ε > δ > | f ( w ) − w | < ε | − w | for all | w − | < δ , w ∈ S ′ . It follows that | I ( z ) | ≤ ε π Z C ( z ) | − w | | w − z | | dw | ≤ ε r ( z ) max w ∈ C ( z ) | − w | = ε r ( z ) (cid:18) + | − z | r ( z ) (cid:19) ≤ ε ( + csc ( β − α )) r ( z ) ≤ ε ( + csc ( β − α )) | − z | . Hence we get f ′ ( z ) = + o (cid:0) | − z | (cid:1) as z → . Also by assumption, we have1 − | f ( z ) | − | z | = − | z | + o (cid:0) | − z | (cid:1) − | z | = + o (cid:0) | − z | (cid:1) as z → . This implies | f ′ ( z ) | − | z | − | f ( z ) | = + o (cid:0) | − z | (cid:1) as z → . (cid:3) R EFERENCES [1] M. Abate, Iterationtheoryofholomorphicmapsontautmanifolds, Mediterranean Press, Rende, 1989.[2] M. Abate, Commonfixedpointsofcommutingholomorphicmaps. Math. Ann. (1989), 645-655.[3] L. V. Ahlfors. AnextensionofSchwarz’slemma. Trans. Am. Math. Soc. (1938), 359–364.[4] G. J. Aladro, SomeconsequencesoftheboundarybehavioroftheCaratheodoryandKoabayashimetricsandapplicationstonormalholomorphicfunctions. Thesis (Ph.D.)?The Pennsylvania State University. 1985. 147pp.[5] L. Baracco, D. Zaitsev, and G. Zampieri. ABurns-Krantztypetheoremfordomainswithcorners.Math. Ann. no. 3 (2006), 491–504.[6] A. F. Beardon. TheSchwarz-Pick Lemmafor derivatives. Proc. Am. Math. Soc., no. 11 (1997), 3255–3256.[7] A.F. Beardon and D. Minda, Amulti-pointSchwarz-Picklemma. J. Anal. Math. (2004), 81-104.[8] A. F. Beardon and D. Minda. The hyperbolic metric and geometric function theory. In Proceedings of theinternational workshop on quasiconformal mappings and their applications, December 27, 2005–January 1,2006, pages 9–56. New Delhi: Narosa Publishing House, 2007.[9] V. Bolotnikov, Auniquenessresultonboundaryinterpolation.Proc. Am. Math. Soc. No. 5 (2008), 1705-1715.[10] F. Bracci, H. Gaussier and A. Zimmer, The geometry of domains with negatively pinched K¨ahler metrics.arXiv:1810.11389.[11] F. Bracci, G. Patrizio, Monge-Amp`ere foliations with singularities at the boundary of strongly convex do-mains. Math. Ann. no. 3 (2005), 499-522. [12] F. Bracci, G. Patrizio, S. Trapani, ThepluricomplexPoissonkernelforstronglyconvexdomains. Trans. Amer.Math. Soc. no. 2 (2009), 979-1005.[13] F. Bracci, D. Zaitsev, Boundaryjetsofholomorphicmapsbetweenstronglypseudoconvexdomains. Journalof Funct. Anal. (2008), 1449-1466.[14] D. M. Burns and S. G. Krantz. RigidityofholomorphicmappingsandanewSchwarzlemmaattheboundary.J. Am. Math. Soc. no. 3 (1994), 661–676.[15] H. Cartan, Les fonctions de deux variables complexes et le probl`eme de la repr´sentation analytique. J. deMath., IX. Ser. (1931), 1–114.[16] C.H. Chang, M.C. Hu, H.P. Lee, Extremalanalyticdiscswithprescribedboundarydata. Trans. Amer. Math.Soc. no. 1 (1988), 355-369.[17] D. Chelst, A generalized Schwarz lemma at the boundary. Proc. Am. Math. Soc. no. 11 (2001), 3275-3278.[18] H. Chen, On the Bloch constant , in: Arakelian, N. (ed.) et al., Approximation, complex analysis, and potentialtheory, Kluwer Academic Publishers (2001), 129–161.[19] K. Diederich, J.-E. Fornæss, Properholomorphicimagesofstrictlypseudoconvexdomains. Math. Ann. no. 2 (1982), 279-286.[20] V.N. Dubinin, The Schwarz inequality on the boundary for functions regular in the disk. J. Math. Sci. no. 6 (2004), 3623-3629.[21] M. Elin, F. Jacobzon, M. Levenshtein, D. Shoikhet,
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