Visibility of Kobayashi geodesics in convex domains and related properties
aa r X i v : . [ m a t h . C V ] J a n VISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINSAND RELATED PROPERTIES
FILIPPO BRACCI † , NIKOLAI NIKOLOV †† , AND PASCAL J. THOMAS Abstract.
Let D ⊂ C n be a bounded convex domain. A pair of distinct boundarypoints { p, q } of D has the visibility property provided there exist a compact subset K p,q ⊂ D and open neighborhoods U p of p and U q of q , such that the real geodesics forthe Kobayashi metric of D which join points in U p and U q intersect K p,q . Every Gromovhyperbolic convex domain enjoys the visibility property for any couple of boundarypoints. The Goldilocks domains introduced by Bharali and Zimmer and the log-typedomains of Liu and Wang also enjoys the visibility property.In this paper we prove that a certain estimate on the growth of the Kobayashi distancenear the boundary points is a necessary condition for visibility and provide new caseswhere this estimate and the visibility property hold.We also exploit visibility for studying the boundary behavior of biholomorphic maps. Contents
1. Introduction 22. Definitions and Preliminaries 43. Gromov hyperbolic domains, visibility and the proof of Theorem 1.3 74. Visibility and growth of the metric 115. Points of infinite type 156. Localization 196.1. The proof of Theorem 6.5. 236.2. The proofs of Theorem 6.7 and Theorem 1.4 246.3. The proof of Proposition 6.9. 25References 25
Mathematics Subject Classification.
Key words and phrases. convex domain, Kobayashi distance, Gromov hyperbolicity, visibility. † Partially supported by PRIN 2017 Real and Complex Manifolds: Topology, Geometry and holo-morphic dynamics, Ref: 2017JZ2SW5, by GNSAGA of INdAM and by the MIUR Excellence De-partment Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUPE83C18000100006. †† Partially supported by the National Science Fund, Bulgaria under contract DN 12/2. Introduction
A bounded domain in D ⊂ C n has the visibility property if the real geodesics for theKobayashi distance k D “bend inside” when connecting points close to the (Euclidean)boundary ∂D (a precise definition is given below).Gromov hyperbolic geodesics space have the visibility property when considering theGromov boundary in the Gromov topology. Therefore, in case ( D, k D ) is Gromov hy-perbolic and the Euclidean boundary ∂D is homeomorphic to the Gromov boundary, D enjoys the visibility property as defined above. In fact, we show that if ( D, k D ) is Gromovhyperbolic, then it has the visibility property if and only if the identity map extends as acontinuous surjective map from the Gromov closure of D to D (see Theorem 3.3).Therefore, in light of [BB], C -smooth bounded strongly pseudoconvex domains enjoythe visibility property. While, by [BGZ], the Euclidean end compactification of a convexdomain which is Gromov hyperbolic with respect to the Kobayashi distance is naturallyhomemorphic to the Gromov compactification. Thus, the visibility property holds forGromov hyperbolic bounded convex domains. It is known by A. Zimmer [Z1] that boundedsmooth convex domains are Gromov hyperbolic with respect to the Kobayashi distance ifand only if they are of finite D’Angelo type, so that finite type smooth bounded convexdomains enjoy the visibility property.In a recent paper, M. Fiacchi [Fia] showed that in C , bounded smooth pseudoconvexdomains of finite D’Angelo type are Gromov hyperbolic and the Gromov boundary ishomeomorphic to the Euclidean boundary, hence, even for those domains the visibilityproperty holds.However, there exist convex domains for which the visibility property holds but theyare not Gromov hyperbolic (see [BZ]). One of the result we prove in this paper is thatsmooth convex domains which are of finite type except at most finitely many points enjoythe visibility property: Theorem 1.1.
Let D ⊂ C n be a bounded convex domain with C ∞ boundary. If all butfinitely many points p ∈ ∂D are of finite D’Angelo type, then D has the visibility property. Then we note that if the visibility property holds in a bounded convex domain, a certainstandard estimate (2.3) of the Kobayashi distance between points tending to distinctboundary points has to be satisfied. In particular, by results of Zimmer [Z2, Propositions3.5 and 4.6], this excludes the presence of analytic discs with nonempty interior in ∂D .By [Z3, Lemma 4.5], the standard estimate (2.3) is satisfied for bounded convex domainswith C boundaries if the boundary points do not share the same (affine) complex tangenthyperplane. We generalize that result and show that (2.3) holds under the same hypothesiseven for non-smooth convex domains. Starting from this, we provide a simple sufficientcondition for visibility, which does not involve any quantitative assumption of contactbetween ∂D and its complex tangent plane: ISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINS 3
Theorem 1.2.
Let D be a bounded convex domain with Dini-smooth boundary such thatany complex line avoiding D meets ∂D in at most one point. Then D has the visibilityproperty. It would be interesting to know whether this theorem remains valid for convex domainswith less smoothness assumption.Note that the presence of a point on the boundary of the domain where the boundary“approaches” very fast a complex tangent line is enough to prevent the first (and crucial)condition in [BZ, Definition 1.1] (see also Definition 2.6 below) to hold. So Theorem 1.2provides new examples of bounded convex domains having the visibility property withoutbeing Goldilocks nor satisfying the weaker hypotheses of [BM, Theorem 1.5].Visibility condition, although weaker than Gromov hyperbolicity, allows to study bound-ary behavior of biholomorphisms (see [BZ]). In Section 3 we explain the underlying phi-losophy in case of Gromov hyperbolic domains, showing that any biholomorphism from abounded domain D to a bounded domain D which has the visibility property extendscontinuously to the boundary provided, for instance, D is either strongly pseudoconvexwith C boundary, or smooth finite type and convex or pseudoconvex and of finite typein C . Next, based on such argument, we localize the result (getting rid of Gromov’shyperbolicity condition) and prove the following: Theorem 1.3.
Let D and D ′ be bounded, complete hyperbolic domains, and assumethat D has a Stein neighborhood basis and D ′ has the visibility property. Suppose thereexists p ∈ ∂D such that ∂D is C -smooth and strongly pseudoconvex at p . If F is abiholomorphism from D to D ′ then F admits non-tangential limit at p . As it follows from the proof, the same result holds provided D is any complete hyperbolicdomain such that it has a geodesic ray γ landing at a point p ∈ ∂D , where ∂D is C -smooth, so that any non-tangential sequence in D converging to p stays at finite hyperbolicdistance from γ .It is possible to forego the global hypothesis of convexity, and find analogues withsuitably localized hypotheses. A point p on the boundary of a domain D is a totally C -convexifiable point if there exists an open neighborhood U of p and a biholomorphismΨ : U → Ψ( U ) so that Ψ( U ∩ D ) is convex and every complex affine line L which containsΨ( p ) and verifies L ∩ Ψ( U ∩ D ) = ∅ has the property that L ∩ Ψ( U ∩ D ) = { p } (seeDefinition 6.6). In Section 6 we prove Theorem 6.7, which has the following consequence: Theorem 1.4.
Let D be a complete hyperbolic bounded domain with Dini-smooth bound-ary and assume all boundary points are totally C -convexifiable. Then D has the visibilityproperty. The plan of the paper is the following. In Section 2 we introduce some preliminarynotations and results. In Section 3 we consider Gromov hyperbolic domains and visibledomains and prove Theorem 1.3. In Section 4 we prove Theorem 1.2. In Section 5, we
F. BRACCI, N. NIKOLOV, AND P. J. THOMAS prove Theorem 1.1 and give a family of examples of smooth bounded convex domainswith (non isolated) boundary points of infinite type which do (and do not) satisfy thevisibility property. Finally, in Section 6, we investigate how to “localize” the previousresults replacing the convex assumption with local conditions near the boundary andprove Theorem 1.4. 2.
Definitions and Preliminaries
In this section, D ⊂ C n is an open connected set. Definition 2.1.
Let x, y, z ∈ D and X ∈ C n . The Kobayashi-Royden (pseudo)metric κ D and the Kobayashi (pseudo)distance k D of D are defined as: κ D ( z ; X ) = inf {| α | : ∃ ϕ ∈ O ( D , D ) : ϕ (0) = z, αϕ ′ (0) = X } , (2.1) k D ( x, y ) = inf γ Z κ D ( γ ( t ); γ ′ ( t )) dt, where the infimum is taken over all piecewise C curves γ : [0 , → D with γ (0) = x and γ (1) = y. A geodesic for k D is a curve σ : I −→ D , where I is an interval in R , such that forany s, t ∈ I , k D ( σ ( s ) , σ ( t )) = | s − t | .If x, y ∈ D , I = [0 , L ] and σ (0) = x, σ ( L ) = y , then L = k D ( x, y ) and we say that σ isa geodesic joining x and y .If I = ( −∞ , + ∞ ) , we say that σ is a geodesic line .If I = [0 , ∞ ) , we say that σ is a geodesic ray . A geodesic ray σ lands if there exists p ∈ D such that lim t →∞ σ ( t ) = p . The domain D is complete hyperbolic if k D is a complete distance, namely, the balls forthe Kobayashi distance are relatively compact. Bounded convex domains are well knownto be complete hyperbolic.If a domain D is complete hyperbolic, by the Hopf-Rinow’s Theorem, ( D, k D ) is ageodesic space, namely, any two points in D can be joined by a geodesic.It is also a fact that when a geodesic exists, it realizes the minimum in (2.1) [Ven,Theorem 3.1]. Definition 2.2.
Let D be complete hyperbolic. Let p, q ∈ ∂D , p = q . We say that thepair { p, q } has visible geodesics if there exist neighborhoods U, V of p, q respectively suchthat U ∩ V = ∅ , and a compact set K ⊂ D such that for any geodesic curve γ : [0 , L ] → D with γ (0) ∈ U , γ (1) ∈ V , then γ ([0 , L ]) ∩ K = ∅ .We say that D has the visibility property if any pair { p, q } ⊂ ∂D , p = q , has visiblegeodesics. ISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINS 5
Note that the pair { p, q } has visible geodesics if and only if there exists a compact set K ⊂ D such that for any sequences ( p k ) k , ( q k ) k ⊂ D , with p k → p , q k → q , then for k large enough any geodesic joining p k to q k intersects K .Our definition of visibility requires that D is complete hyperbolic and it is slightlydifferent than the definition in [BZ], [BM], where the authors need to consider the class of“quasi-geodesics” for dealing with not necessarily complete hyperbolic domains. In caseof complete hyperbolic domains, from a practical point of view, the two definitions arehowever essentially equivalent.For a point z ∈ D , we let δ D ( z ) := inf {k z − w k : w ∈ C n \ D } , the (Euclidean) distance of z from the boundary of D Since the Kobayashi distance is always larger than the Carath´eodory distance, forconvex domains D , the estimate in [Nik, (2)] (coming from the proof of [Blo, Theorem5.4]) yields that for x, y ∈ D ,(2.2) k D ( x, y ) ≥ (cid:12)(cid:12)(cid:12)(cid:12) log δ D ( y ) δ D ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . The previous estimate is pretty good when x goes to the boundary and y stays compactlyin D . However, when both x, y go to the boundary, it does not give much information. Infact, it is known in many cases that when x and y converge to different boundary points,the Kobayashi distance between the two explodes like − / D is strongly (pseudo)convex with C boundary,see, e.g. , [Aba, Corollary 2.3.55]). We thus give the following definition: Definition 2.3.
Let p, q ∈ ∂D , p = q . We say that the pair { p, q } verifies the standardestimate if (2.3) lim inf ( x,y ) → ( p,q ) [2 k D ( x, y ) + log δ D ( x ) + log δ D ( y )] > −∞ . We make a quick observation.
Proposition 2.4.
Let D be a bounded convex domain. If the pair { p, q } has visiblegeodesics, then it satisfies the standard estimate.Proof. Indeed, take
U, V, K as in Definition 2.2 and set δ D ( K ) := min z ∈ K δ D ( z ) >
0. Forany x ∈ D ∩ U, y ∈ D ∩ V there exists a geodesic γ joining x to y since D is completehyperbolic. By visibility, γ intersects K in a point z . Hence, k D ( x, y ) = k D ( x, z )+ k D ( z, y ).Applying (2.2) twice, we have2 k D ( x, y ) ≥ − log δ D ( x ) − log δ D ( y ) + 2 log δ D ( K ) , and we are done. (cid:3) F. BRACCI, N. NIKOLOV, AND P. J. THOMAS
The standard estimate is a lower estimate for the growth of the Kobayashi distancenear the boundary of D .We fix some terminology to describe analogous upper estimates. Definition 2.5.
We say that:(1) D has α -growth if there exist some α > , β > and x ∈ D such that sup z ∈ D (cid:0) k D ( x , z ) − βδ D ( z ) − α (cid:1) < ∞ . (2) D has α -log-growth if there exist some α > and x ∈ D such that sup z ∈ D ( k D ( x , z ) − α log δ D ( z )) < ∞ . Note that in particular when ∂D is Lipschitz, D has α -log-growth [BZ, Lemma 2.3]. Inparticular convex domains automatically satisfy that property.Moreover, recall that a point p ∈ ∂D is a Dini-smooth point if ∂D is C -smooth at p and the inner unit normal vector ν q is a Dini-continuous function for q close to p , namely, Z ω ( t ) t dt < + ∞ , where ω ( t ) := sup {k ν x − ν y k : k x − y k < t, x, y ∈ ∂D } . The boundary ∂D is Dini-smooth if all its points are Dini-smooth points.If ∂D is Dini-smooth, then D has -log-growth by [NA, Corollary 8]. However, thereexist C -smooth (but not Dini-smooth) domains for which D has not -log-growth [NPT,Example 2].Finally, we recall a definition from [BZ]. Definition 2.6. [BZ, Definition 1.1]
Let M D ( r ) := sup (cid:26) κ D ( x ; X ) : δ D ( x ) ≤ r, k X k = 1 (cid:27) . A bounded domain D ⊂ C n is a Goldilocks domain if(1) for some ε > we have Z ε M D ( r ) r dr < ∞ , (2) D has α -log-growth for some α > . The first property is satisfied, for example, for any bounded pseudoconvex domain offinite type (in sense of D’Angelo) [BZ, Lemma 2.6], but the class of domains with theGoldilocks property strictly contains the class of domains of finite type.It is proved in [BZ, Theorem 1.4] that a Goldilocks domain has an extended visibilityproperty which implies in particular the one in Definition 2.2 when the Kobayashi dis-tance is geodesic. A more general result implying visibility is given in [BM, Theorem 1.5
ISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINS 7 (General Visibility Lemma)], which essentially reduces to the Goldilocks case when thedomain is convex.
Definition 2.7. A complex face of a convex domain D is L ∩ ∂D , where L an affinecomplex line such that L ∩ D = ∅ and L ∩ ∂D = ∅ .Given a point p ∈ ∂D , the multiface F p of p is the union of all the complex faces of D which contain p . Note that the condition in Theorem 1.2 means that F p = { p } for any p ∈ ∂D .Let L be an affine complex line such that L ∩ D = ∅ and p ∈ L ∩ ∂D . Since L and D are convex and disjoint and D open, by Hahn-Banach’s extension theorem, there existsa vector v ∈ C n such that Re h z − p, v i < z ∈ D and Re h z − p, v i ≥ z ∈ L (here h· , ·i denotes the usual Hermitian product in C n ). Actually, since L is acomplex affine line it turns out that h z − p, v i = 0 for all z ∈ L . This implies that L is contained in the complex affine hyperplane H := { z ∈ C n : h z − p, v i = 0 } . Such anhyperplane H is called a complex supporting hyperplane of D at p (see [AR, Def. 3]).The intersection of D with all complex supporting hyperplanes of D at p is denoted byCh( p ). In [AR], the point p is called a strictly C -linearly convex point if Ch( p ) = { p } .By the previous remark, a point for which the multiface F p = { p } is a strictly C -linearlyconvex point. The converse is not true in general: the multiface of the bidisc D × D at(1 ,
1) is ( D × { } ) ∪ ( { } × D ), while Ch((1 , { (1 , } .3. Gromov hyperbolic domains, visibility and the proof of Theorem 1.3
The aim of this section is to see how visibility is related to Gromov visibility in boundeddomain for which the Kobayashi distance is complete and Gromov hyperbolic.We briefly recall the definition of Gromov compactification for a complete hyperbolicdomain D such that ( D, k D ) is Gromov hyperbolic. Let z ∈ D . Let Γ z be the set of allgeodesic rays γ such that γ (0) = z . Two geodesic rays γ, η ∈ Γ z are equivalent ifsup t ≥ k D ( γ ( t ) , η ( t )) < + ∞ . The set of equivalence classes in Γ z is the Gromov boundary ∂ G D . A sequence { σ k } ⊂ ∂ G D converges to σ ∈ ∂ G D if there exist representative γ k ∈ σ k such that { γ k } convergesuniformly on compacta to a geodesic ray γ ∈ σ . Also, a sequence { z k } ⊂ D convergesto σ ∈ ∂ G D if, given geodesics γ k : [0 , R k ] → D such that γ k (0) = z and γ k ( R k ) = z k ,then { γ k } converges uniformly on compacta to a geodesic ray γ ∈ σ . We denote by D G := D ∪ ∂ G D the Gromov compactification of D with the topology introduced above.We start with the following simple result: Lemma 3.1.
Let D be a bounded, complete hyperbolic domain with the visibility property.If { z k } , { w k } ⊂ D converge to different points on the boundary of D , then k D ( z k , w k ) →∞ . F. BRACCI, N. NIKOLOV, AND P. J. THOMAS
Proof.
Consider a geodesic γ k joining z k to w k . By the visibility hypothesis, there exista compact set K ⊂ D and t k such that γ k ( t k ) ∈ K for all k . Since k D is complete,this implies that k D ( z k , γ k ( t k )) and k D ( w k , γ k ( t k )) → ∞ , thus since γ k is a geodesic, k D ( z k , w k ) = k D ( z k , γ k ( t k )) + k D ( w k , γ k ( t k )) → ∞ . (cid:3) Lemma 3.2.
Let D be a bounded, complete hyperbolic domain with the visibility property.Then any geodesic ray lands at a point on ∂D .Conversely, let z ∈ D . Let { z k } ⊂ D be a sequence which converges to a point p ∈ ∂D , and let γ k be geodesics joining z to z k . Then, up to subsequences, { γ k } convergesuniformly on compacta to a geodesic ray landing at p .Proof. Let γ be a geodesic ray in D . Since D is bounded and complete hyperbolic, thecluster set Γ of γ at + ∞ is contained in ∂D .Suppose there are two distinct points p, q ∈ Γ, and sequences s k → ∞ and t k → ∞ suchthat lim k →∞ γ ( s k ) = p , lim k →∞ γ ( t k ) = q . Passing to subsequences if necessary, we mayassume s k < t k < s k +1 for all k . Then by the visibility property applied to the geodesics γ | [ s k ,t k ] , there is a compact set K ⊂ D such that for each k , there exists s ′ k ∈ ( s k , t k ) suchthat we have γ ( s ′ k ) ∈ K . But then Γ ∩ K = ∅ , a contradiction.Conversely, fix z ∈ D and a sequence { z k } converging to p . We can assume that k D ( z , k k ) is strictly increasing. Let γ k be a geodesic joining z to z k . By Kobayashicompleteness of D , for any fixed m and k ≥ m , the images of γ k | [0 ,k D ( z ,z m )] are containedin a fixed compact set K m . There exists c > z ∈ K m , X ∈ C n , c k X k ≤ κ D ( z, X ) ≤ c − k X k , so the family ( γ k | [0 ,k D ( z ,z m )] ) k ≥ m is equicontinuous withrespect to the Euclidean distance. By Arzel`a-Ascoli’s theorem, and a diagonal argument,up to subsequences, we can assume that ( γ k ) is uniformly convergent on compacta to ageodesic ray γ . For what we already proved γ lands at some point q ∈ ∂D . We haveto show that q = p . If this is not the case, however, we can find sequences of positivereal numbers { s k } and { t k } converging to + ∞ , and we can assume s k < t k , such that γ k ( s k ) → q and γ k ( t k ) → p . But then, by the visibility property, γ k | [ s k ,t k ] has to intersecta given compact set K ⊂ D for all k , say, γ k ( t ′ k ) ∈ K for some t ′ k ∈ [ s k , t k ]. But, t ′ k = k D ( z , γ k ( t ′ k )) ≤ max z ∈ K k D ( z , z ) < + ∞ , a contradiction. Hence p = q and we are done. (cid:3) The visibility property in Gromov hyperbolic spaces has a strong consequence. As amatter of notation, we say that a geodesic line γ : ( −∞ , + ∞ ) → D is a geodesic loop if γ has the same cluster set at + ∞ and −∞ .It is easy to see that, if ( D, k D ) is Gromov hyperbolic and γ is a geodesic line, and ifwe let σ ± := lim t →±∞ γ ( t ) ∈ ∂ G D (the limit understood in the Gromov topology), then σ + = σ − . Thus, in this sense, D G has no geodesic loops. Theorem 3.3.
Let D be a complete hyperbolic bounded domain. Assume ( D, k D ) isGromov hyperbolic. Then D has the visibility property if and only if the identity map ISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINS 9 extends as a continuous surjective map
Φ : D G → D . Moreover, Φ is a homeomorphismif and only if D has no geodesic loops.Proof. If the identity map extends as a surjective continuous map from D G to D , since( D, k D ) is Gromov hyperbolic and hence D G enjoys the Gromov visibility property, itfollows at once that D has the visibility property. Moreover, if the extension is a homeo-morphism, then D has no geodesic loops because D G has none.Conversely, assume that D has the visibility property. We first define a map Φ : D G → D as follows: if z ∈ D then Φ( z ) = z . If σ ∈ ∂ G D , let γ ∈ σ be such that γ ∈ Γ z . Thenby Lemma 3.2, γ lands at some point p ∈ ∂p . Note that by Lemma 3.1, the point p doesnot depend on the representative γ chosen. Therefore, we can set Φ( σ ) := p .By Lemma 3.2, the map Φ is surjective and Φ( z k ) → Φ( σ ) if { z k } ⊂ D is a sequencewhich converges in the Gromov topology to σ ∈ ∂ G D . An argument similar to the one inthe proof of Lemma 3.2 shows that Φ( σ k ) → Φ( σ ) if { σ k } ⊂ ∂ G D converges to σ in theGromov topology.Assume now that D has no geodesic loops. Since D G , D are Hausdorff and compact, ifwe show that Φ is injective, then it is also a homeomorphism. Assume by contradictionthat Φ( σ ) = Φ( θ ) for some σ, θ ∈ ∂ G D , σ = θ . Let γ ∈ σ and η ∈ θ be two representative.Hence, there exists a sequence { t k } converging to + ∞ such thatlim k →∞ k D ( γ ( t k ) , η ( t k )) = + ∞ . We first claim that lim k →∞ inf s ≥ k D ( η ( s ) , γ ( t k )) = + ∞ . Indeed, assume by contradiction that this is not the case. Then there exists a sequence { s k } of positive numbers and R > kk D ( η ( s k ) , γ ( t k )) = min s ≥ k D ( η ( s ) , γ ( t k )) ≤ R. Since | t k − s k | = | k D ( z , γ ( t k )) − k D ( z , η ( s k )) | ≤ k D ( η ( s k ) , γ ( t k )) ≤ R, we have for all k , k D ( η ( t k ) , γ ( t k )) ≤ k D ( η ( s k ) , γ ( t k )) + k D ( η ( s k ) , η ( t k )) ≤ R, a contradiction. Therefore the claim holds.Now, let β k : [0 , T k ] → D be a geodesic joining γ ( t k ) and η ( t k ). Since ( D, k D ) is Gromovhyperbolic, it follows that there exists M > k and all m ≤ k ,min { k D ( γ ( t m ) , η ([0 , t k ])) , k D ( γ ( t m ) , β k ([0 , T k ])) } ≤ M. By the claim, if m is sufficiently large, the only possibility is that k D ( γ ( t m ) , β k ([0 , T k ])) ≤ M . This implies that there exists a compact set K ⊂ D such that β k intersects K forall k . We can reparametrize β k in such a way that β k : [ − s k , r k ] → D and β k (0) ∈ K . Using such a parametrization and arguing as in the proof of Lemma 3.2, we see that byvisibility, β k converges to a geodesic line β : ( −∞ , + ∞ ) → D and that lim t →±∞ β ( t ) = p .Namely, β is a geodesic loop, contradicting our hypothesis. (cid:3) In dimension one, we have a simple characterization of visible simply connected do-mains:
Corollary 3.4.
Let D ⊂ C be a bounded simply connected domain. Then D has thevisibility property if and only if ∂D is locally connected.Proof. Assume ∂D is locally connected. Let f : D → D be a Riemann map. Since ( D , k D )is Gromov hyperbolic, so is ( D, k D ). Since ∂D is locally connected, it follows by theCarath´eodory extension theorem (see, e.g. [BCD, Thm. 4.3.1]) that f extends continu-ously to the boundary, call ˜ f such an extension. Now, f − extends as a homeomorphism–denote it by ˆ f − —from D G to D G and the identity map id D extends as a homeomorphismˆ id D from D G to D . Therefore, ˆ id D := ˜ f ◦ ˆ id D ◦ ˆ f − is a continuous surjective extension of id D . Thus, by Theorem 3.3, D has the visibility property.Conversely, if D has the visibility property, then by Theorem 3.3, id D extends as asurjective continuous map from D G to D . Since ∂ G D is homeomorphic to ∂ G D and thelatter is homeomorphic to ∂ D , it follows that there exists a continuous surjective functionfrom ∂ D to ∂D , and hence ∂D is locally connected (see, e.g. [BCD, Thm. 4.3.1]). (cid:3) The following example provides a domain with the visibility property and geodesicloops:
Example 3.5.
The domain D := D \ { [0 , } is simply connected and ∂D is locallyconnected, thus, D has the visibility property. However, D has geodesic loops: take ξ , ξ ∈ ∂ D , ξ = ξ such that f ( ξ ) = f ( ξ ) = 1 / (see, e.g. [BCD, Prop. 4.3.5] ), let γ be thegeodesic in D whose closure contains ξ and ξ . Hence, f ( γ ) is a geodesic loop in D . Since any biholomorphism between two Gromov hyperbolic domains extends as a home-omorphim between the Gromov closure of the domains, we have also the following directconsequence:
Proposition 3.6.
Let D , D ⊂ C n be complete hyperbolic bounded domains. Assume D , D have the visibility property. If ( D , k D ) is Gromov hyperbolic and D has nogeodesic loops then every biholomorphism F : D → D extends continuously on ∂D . In particular the previous proposition applies when D is strongly pseudoconvex, orconvex and Gromov hyperbolic (so, for instance if ∂D is smooth and of finite type), or D ⊂ C is smooth, pseudoconvex and of finite type.One can localize the previous argument, by getting rid of Gromov hyperbolicity con-dition, provided one has some knowledge on geodesic rays landing at a given boundarypoint. This is the content of Theorem 1.3, which we are now going to prove: ISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINS 11
Proof of Theorem 1.3.
By [BFW, Thm. 2.6] (which is actually based on [DFW, Thm 1.1]and allows to replace the hypothesis that D is strongly pseudoconvex in [BFW, Thm. 2.6]with D having a Stein neighborhood basis), there exist a C -smooth bounded stronglyconvex domain W ⊂ C n and a univalent map Φ : D → C n such that Φ extends C upto D , Φ( p ) ∈ ∂W , Φ( D ) ⊂ W and there exists an open neighborhood U of Φ( p ) suchthat W ∩ U = Φ( D ) ∩ U . By [BST, Lemma 4.5], there exists a complex geodesic for W ( i.e. , an isometry between k D and k W ) ϕ : D → W such that ϕ ( D ) ⊂ U , ϕ (1) = p and ϕ is also a complex geodesic for Φ( D ). Note that [0 , ∋ r ϕ ( r ) is (once suitablyreparametrized in hyperbolic arc-length) a geodesic ray in Φ( D ), which lands at p and itis transverse to ∂D (by Hopf’s Lemma). It follows that D has a geodesic ray γ landingat p non-tangentially.Since F ( γ ) is a geodesic ray in D ′ , by Lemma 3.2, it lands at some point q ∈ ∂D ′ .In order to prove the theorem, we need to show that if { z k } ⊂ D is a sequence convergingnon-tangentially to p then { F ( z k ) } converges to q .In order to see this, fix a sequence { z k } ⊂ D converging non-tangentially to p . Let B ⊂ D be a ball tangent to ∂D at p . Since k D | B ≤ k B , it follows from [BF, Lemma 2.3] that { z k } stays at finite Kobayashi distance from γ . Hence, there exist a sequence of positivereal numbers { t k } converging to + ∞ and a constant C > k D ( γ ( t k ) , z k ) ≤ C for all k . Thus, k D ′ ( F ( z k ) , F ( γ ( t k )) ≤ C for all k , and, by Lemma 3.1 it follows that { F ( z k ) } converges to q . (cid:3) Visibility and growth of the metric
Our first result generalizes [Z3, Lemma 4.5] to the case of domains with irregular bound-aries. In this case, some boundary points do not have a well defined tangent hyperplane,and then the notion of multiface is needed.We start with the following result:
Theorem 4.1.
Let D be a bounded convex domain, and p, q ∈ ∂D such that F p ∩ F q = ∅ .Then the pair { p, q } verifies the standard estimate.Proof. Let { x k } ⊂ D be a sequence converging to p and { y k } ⊂ D a sequence converg-ing to q . For each k , choose a point p k such that k x k − p k k = δ D ( x k ). Then, because B ( x k , δ D ( x k )) ⊂ D and p ∈ B ( x k , δ D ( x k )), there exists a unique complex tangent hyper-plane H p k to ∂D at p k . Up to passing to subsequences, we can assume that H p k converges(in the Hausdorff distance when restricted to a fixed ball) to H p , a complex supportinghyperplane for D at p . Then H p ∩ D ⊂ F p . In the same way, we get H q , a complexsupporting hyperplane for D at q .Since F p ∩ F q = ∅ , it follows that ( H p ∩ H q ) ∩ D = ∅ and, D being bounded, theEuclidean distance of H p ∩ D from H q ∩ D is positive. We write, for η > N p ( η ) := { z ∈ C n : dist( z, H p ) ≤ η } , N q ( η ) := { z ∈ C n : dist( z, H q ) ≤ η } . Fix η > N p ( η ) ∩ N q ( η ) stays at a positive distance from D . Let N p k ( η ) := { z ∈ C n : dist( z, H p k ) ≤ η } , N q k ( η ) := { z ∈ C n : dist( z, H q k ) ≤ η } . Then N p k ( η ) ∩ D converges to N p ( η ) ∩ D in the Hausdorff distance (and the same holdsfor q k and q respectively). So for k large enough,( N p k ( η ) ∩ D ) ∩ ( N q k ( η ) ∩ D ) = ∅ . Suppose now that k is large enough so that δ D ( x k ) , δ D ( y k ) < η . The affine real tangenthyperplane to ∂D at p k , T p k , is orthogonal to the real line passing through x k and p k andcontains the affine complex hyperplane H p k .Let π ′ k be the orthogonal projection to the complex line through x k and p k , parallel to H p k . Take a complex coordinate on this line so that p k is represented by 0 and the inwardhalf line from p k containing x k goes to the positive imaginary half axis. Then π ′ k ( N p k ( η )) = D (0 , η ), π ′ k ( x k ) = iδ D ( x k ), and π ′ k ( D ) ⊂ { Im z > } =: H . Since holomorphic mapscontract the Kobayashi distance,(4.2) k D ( x k , D \ N p k ( η )) ≥ k H (cid:0) iδ D ( x k ) , H \ D (0 , η ) (cid:1) = 12 log 1 δ D ( x k ) −
12 log 1 η , the last equality being obtained by an explicit computation. An analogous inequality canbe proved for k D ( y k , D \ N q k ( η )) by using an orthogonal projection π ′′ k parallel to H q k .Finally, any curve from x k to y k must contain a point z k ∈ F k := D \ ( N p k ( η ) ∪ N q k ( η )).By the above estimates, k D ( x k , y k ) ≥ inf z ∈ F k k D ( x k , z ) + inf z ∈ F k k D ( z, y k ) ≥
12 log 1 δ D ( x k ) + 12 log 1 δ D ( y k ) − log 1 η , and we are done. (cid:3) Corollary 4.2. If F p = { p } , then any pair { p, q } verifies the standard estimate. Theorem 4.3.
Let D be a complete hyperbolic bounded domain with -log-growth. Then D has the visibility property if and only if any pair of points { p, q } ⊂ ∂D satisfies thestandard estimate.Proof. One implication follows from Proposition 2.4.Now suppose any pair of points in ∂D satisfies the standard estimate. Let p = q ∈ ∂D .We argue by contradiction. Then, we can find p k → p , q k → q , and γ k geodesics with γ k (0) = p k , γ k ( L k ) = q k , such that for any K compact subset of D , there exists k suchthat for k ≥ k , γ k ([0 , L k ]) ∩ K = ∅ .For k large enough, there exists w k := γ k ( t k ) such that k p k − γ k ( t k ) k = k q k − γ k ( t k ) k .Passing to a subsequence if needed, we may assume that w k → w ∈ D , with k w − p k = k w − q k , and in particular w = p, w = q . Since any compact subset of D is avoided by ISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINS 13 γ k for k large enough, w ∈ ∂D . Applying the standard estimate to the pairs { p, w } and { w, q } , we have k D ( p k , q k ) = k D ( p k , w k ) + k D ( w k , q k ) ≥ C p,w + C w,q −
12 log δ D ( p k ) −
12 log δ D ( q k ) − log δ D ( w k ) . On the other hand, the -log-growth property implies that k D ( p k , q k ) ≤ k D ( p k , z ) + k D ( z , q k ) ≤ C −
12 log δ D ( p k ) −
12 log δ D ( q k ) , which yields a contradiction. (cid:3) Now we are in good shape to prove Theorem 1.2:
Proof of Theorem 1.2.
By [NA, Corollary 8] the domain D has -log-growth. Since F p = { p } for any p ∈ ∂D by the hypothesis, Theorem 4.1 shows that any pair of distinct points { p, q } in ∂D verifies the standard estimate. Thus Theorem 1.2 follows from Theorem 4.3. (cid:3) One may ask whether a full converse of Proposition 2.4 holds. It does not, and it seemsthat having complex faces on the boundary “on the way” from p to q is an obstruction. Proposition 4.4.
Let D := D , p := ( − , , q := (1 , . Then the pair { p, q } satisfiesthe standard estimate, but not the visibility property.Proof. Let p ′ → p , q ′ → q . If p ′ , q ′ are close enough to p and q respectively, δ D ( p ′ ) = 1 −| p ′ | , δ D ( q ′ ) = 1 − | q ′ | , and using the projection on D × { } , k D ( p ′ , q ′ ) ≥ k D ( p ′ , q ′ ) = − log(1 −| p ′ | ) − log(1 − | q ′ | ) + O (1). So the standard estimate holds.Consider the points p ε := ( − ε, q := (1 − ε, p ε to q ε is given by the line segment [ − ε, − ε ] × { } . But another one can be obtainedin the following way: let c > k D (0 , − ε / ) < k D (1 − ε, − cε / )(this is possible since k D (0 , − ε / ) = − log ε + O (1) = k D (1 − ε, − cε / ) + O (1)). Let˜ p ε := ( − cε / , − ε / ), ˜ q ε := (1 − cε / , − ε / ). Let γ ε be the curve made up of ageodesic from p ε to ˜ p ε , followed by the geodesic from ˜ p ε to ˜ q ε (a horizontal line segment),followed by a geodesic from ˜ q ε to q ε . Since k D ( z, w ) = max( k D ( z , w ) , k D ( z , w )), we seethat ℓ ( γ ε ) = k D ( − ε, − cε / ) + k D ( − cε / , − cε / ) + k D (1 − cε / , − ε )= k D ( − ε, − ε ) , so γ ε is a geodesic segment too. But γ ε ⊂ D \ D (0 , − cε / ) avoids any compact set inthe bidisc as ε →
0, so the pair { p, q } does not have the visibility property. (cid:3) It remains an open question whether one can find such an example with ∂D smooth.One might ask whether the hypothesis F p ∩ F q = ∅ of Theorem 4.1 can be weakened.A natural weaker condition is to assume that p F q (which implies that also q F p ).Under such a condition we can prove: Proposition 4.5.
Let D be a convex domain (not necessarily bounded), and let p, q ∈ ∂D be such that q F p . Then lim inf x → p,y → q (2 k D ( x, y ) + max { log δ D ( x ) , log δ D ( y ) } ) > −∞ . Proof.
By definition of multiface, if q F q then also p F q .For a point z ∈ D , and a vector v ∈ C n , we let(4.3) δ D ( z ; v ) := sup { r > z + λv ∈ D ∀ λ ∈ C , | λ | < r } . It follows by the estimate [NT, p. 633] that2 k D ( x, y ) ≥ log { log δ D ( x ; x − y k x − y k ) , log δ D ( y ; x − y k x − y k ) } ! . We claim that there is a constant c so that δ D ( x ; x − y k x − y k ) ≤ cδ D ( x ) and δ D ( y ; x − y ) ≤ cδ D ( y ),for x close enough to p and y close to q , from which the statement follows at once.In order to prove the claim, suppose by contradiction that there exist x k → p , y k → q ,such that(4.4) δ D ( x k ) = o ( δ D ( x k ; x k − y k k x k − y k k )) . Then v k := x k − y k k x k − y k k → v := p − q k p − q k . Let p k ∈ ∂D be such that k x k − p k k = δ D ( x k ),and let H k be the supporting complex hyperplane at p k —note that it is unique because B ( x k , δ D ( x k )) ⊂ D and p k ∈ ∂B ( x k , δ D ( x k )). Passing to a subsequence if needed, wemay assume that H k converges to a supporting complex hyperplane H ∞ at p . Then,decomposing the vectors v k in components parallel and orthogonal to H k , (4.4) impliesthat v ∈ H ∞ . This means that the direction p − q is parallel to F p , and by convexity, that q ∈ F p , contradicting the hypothesis.A similar argument shows that δ D ( y ; x − y ) ≤ cδ D ( y ), for y close to q , and we aredone. (cid:3) The conclusion of the previous proposition cannot be improved: consider the bidisc D := D , and p := (1 , q := (0 , x := (1 − ε, y := (0 , − ε ). Then q / ∈ F p , p / ∈ F q ,but F p ∩ F q = { (1 , } and k D ( x, y ) = 12 log 2 − εε ≤
12 log 1 δ D ( x ) + log 2 = 12 log 1 δ D ( y ) + log 2 . ISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINS 15 Points of infinite type
By [Z1] bounded smooth convex finite D’Angelo type domains are Gromov hyperbolicand so they have the visibility property by [BGZ]. In this section we first prove Theo-rem 1.1, proving that, although not Gromov hyperbolic, smooth bounded convex domainswhose boundary points are of finite type except at most finitely many have the visibilityproperty.To this aim, we need to set up some notation and preliminary notions.In this section, D ⊂ C n is a bounded convex domain with smooth boundary.Since any bounded domain with C boundary satisfies the inner ball condition, thereexists δ c > z ∈ D with δ D ( z ) < δ c , there exists a unique point ζ ∈ ∂D such that k z − ζ k = δ D ( z ). We denote ζ = π ( z ). The fiber π − ( π ( z )) is a subset of thereal normal line to ∂D at π ( z ). The map z ( π ( z ) , δ D ( z )) is a diffeomorphism from { z ∈ D : δ D ( z ) < δ c } to ∂D × (0 , δ c ).Using the inner ball condition and an affine mapping, and taking δ c smaller as needed,it is easy to see that there exists a constant A such that for all p, q ∈ D such that π ( p ) = π ( q ) and δ c ≥ δ D ( p ) ≥ δ D ( q ), then(5.1) k D ( p, q ) ≤
12 log δ D ( p ) δ D ( q ) + A . Note also that K c := { z ∈ D : δ D ( z ) ≥ δ c } is a compact set, and so bounded in theKobayashi distance of D .We recall the definition of a minimal basis at a point z , and of the directional distancesto the boundary τ j ( z ), as given in [NT].Let z ∈ D and let z := π ( z ) and τ ( z ) := k z − z k = δ D ( z ) . Let H = z + ( C ( z − z )) ⊥ ,where, for a complex space V ⊂ C n we let V ⊥ denotes the complex space orthogonal(with respect to the standard Hermitian product) to V . Let D = D ∩ H . Note that D is a convex domain in the affine complex space H of dimension n − z ∈ D .Let z ∈ ∂D so that τ ( z ) := k z − z k = δ D ( z ) . Let H = z + (span C ( z − z, z − z )) ⊥ ,D = D ∩ H and iterate the construction for n steps. Thus we get an orthonormalbasis of the vectors e j = z j − z k z j − z k , ≤ j ≤ n, which is called minimal for D at z , andpositive numbers τ ( z ) ≤ τ ( z ) ≤ · · · ≤ τ n ( z ) (the basis and the numbers are not uniquelydetermined).From the last formula in the proof of [Gau, Lemma 1.3] (p. 379, line 4) we have Lemma 5.1.
Let S ⊂ ∂D be compact and let U ⊂ ∂D be a neighborhood of S . Supposethere exists M > such that the type of ζ ∈ U is bounded from above by M . Then thereexist a neighborhood U ⊆ U of S , ˜ δ > and C > such that for every z ∈ D with δ D ( z ) ≤ ˜ δ and π ( z ) ∈ U , we have τ j ( z ) ≤ C δ D ( z ) /M for all ≤ j ≤ n . The following lemma is an immediate consequence of [NT, Theorem 1, (ii)].
Lemma 5.2. If q, z ∈ D are such that k D ( q, z ) < r , then max ≤ j ≤ n | z j − q j | τ j ( q ) < e r − , where ( z , . . . , z n ) and ( q , . . . , q n ) are the coordinates of z and q in the affine coordi-nates system { O ; e , . . . , e n } , where O is the origin in C n and { e , . . . , e n } is a minimalorthonormal basis for D at q . We now prove that if two points with comparable distance from the boundary are joinedby a geodesic that stays as close to the boundary as they are, then they must be close toeach other in the Euclidean distance. This is achieved by comparing the Kobayashi lengthof a curve staying close enough to the boundary with an upper bound for the Kobayashidistance between its extremities.
Lemma 5.3.
Let S ⊂ ∂D be compact and let U ⊂ ∂D be a neighborhood of S . Supposethere exists M > such that the type of ζ ∈ U is bounded from above by M . Let δ ∈ (0 , min { δ c , e − } ) . Suppose p, q ∈ D and δ D ( p ) , δ D ( q ) ∈ (0 , δ ) . Let γ be a geodesicjoining p and q such that δ D ( γ ( t )) ≤ δ and π ( γ ( t )) ∈ S for any t . Then there existconstant A , C > , depending only on S and D , such that k p − q k ≤ C δ /M (cid:18) log 1 δ + A (cid:19) . Proof.
In what follows, C stands for a constant whose value may change from line to line.Consider two points z, w such that π ( z ) , π ( w ) ∈ S and k D ( z, w ) ≤
1. Then Lemma 5.2implies that | z j − w j | ≤ ( e − τ j ( z ) for 1 ≤ j ≤ n (where ( z , . . . , z n ) and ( w , . . . , w n )are the coordinates of z and w in a minimal orthonormal basis for D at z ).Then, by Lemma 5.1,(5.2) k z − w k ≤ √ n max ≤ j ≤ n | z j − w j | ≤ C max ≤ j ≤ n τ j ( z ) ≤ Cδ D ( z ) /M < Cδ /M . In particular, if k D ( p, q ) ≤
1, the result holds whenever C ≥ C and A ≥ δ > δ < e − ).We can thus assume that k D ( p, q ) >
1. Write k D ( p, q ) = m + s , with m ∈ N , m ≥ s ∈ [0 , γ (0) = p and γ ( m + s ) = q . In particular, the hyperbolic length ℓ ( γ ) of γ is greater than or equal to m . Since k D ( γ ( k + 1) , γ ( k )) = 1 for k = 0 , . . . , m − k D ( γ ( m ) , γ ( m + s )) ≤
1, by (5.2), we have for k = 0 , . . . , m − k γ ( k + 1) − γ ( k ) k < Cδ /M , and k γ ( m + s ) − γ ( m ) k < Cδ /M . By the triangle inequality,(5.3) k p − q k < ( m + 1) Cδ /M ≤ ( ℓ ( γ ) + 1) Cδ /M . ISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINS 17
Now, let p , q ∈ D be such that π ( p ) = π ( p ) and π ( q ) = π ( q ) and δ D ( p ) = δ D ( q ) = δ c . Since γ is a geodesic, and using (5.1), we have ℓ ( γ ) ≤ k D ( p, p ) + k D ( p , q ) + k D ( q , q ) ≤ log δ c δ + 2 A + diam K c , where diam K c denotes the diameter in the Kobayashi distance of K c , which is finite.Putting together the previous inequality and (5.3) we have the result. (cid:3) Now we are in a position to prove Theorem 1.1.
Proof of Theorem 1.1.
Let p, q ∈ ∂D . We aim to show that there exists δ p,q > { p k } , { q k } ⊂ D converging to p and q respectively, for k large enough, anygeodesic from p k to q k intersects { z ∈ D : δ D ( z ) ≥ δ p,q } .We argue by contradiction and suppose there is no such δ p,q . Let γ k be a geodesic suchthat γ k (0) = p k and γ k ( R k ) = q k for some R k >
0. Then, for any δ > k δ sothat δ D ( γ k ( t )) ≤ δ for any t and k ≥ k δ . Case 1.
Either q or p (or both) is of finite type.Without loss of generality, we assume q is of finite type. Then, for r small enough,any point in U := B ( q, r ) ∩ ∂D has type bounded from above by some M >
0. Let S := B ( q, r ) ∩ ∂D . We can take r so small that p / ∈ U . Let U := π − ( S ) ⊂ D \ K c . For k large enough, q k ∈ U and γ k ([0 , R k ]) ¯ U . Let t kp := sup { t ∈ [0 , R k ) : γ k ( t ) / ∈ U } and let p ′ k := γ k ( t kp ). Then γ k | [ t kp ,R k ] is a geodesic from p ′ k to q k such that k p ′ k − q k k ≥ r/ π ( γ k ( t )) ∈ S and δ D ( γ k ( t )) ≤ δ for all t ∈ [ t kp , R k ]. In particular, if δ < min { δ c , e − } , byLemma 5.3 we have for all k , r ≤ k p ′ k − q k k ≤ C δ /M (cid:18) log 1 δ + A (cid:19) , which provides a contradiction for δ → Case 2. p and q are of infinite type.Let E ⊂ ∂D be the set of points of infinite type different from p and q . By hypothesis, E is finite. For r > E r := { z ∈ C n : dist( z, E ) < r } .Let r > B ( p, r ) ∩ B ( q, r ) = ∅ and ( B ( p, r ) ∪ B ( q, r )) ∩ E r = ∅ . Then U := ∂D \ (cid:16) B ( p, r/ ∪ B ( q, r/ ∪ E r/ (cid:17) is an open set in ∂D formed by points of typeat most M for some M > S := ∂D \ ( B ( p, r ) ∪ B ( q, r ) ∪ E r ). Then S ⊂ U is a compact set.Now we can argue by contradiction as before. Given two sequences { p k } and { q k } in D converging to p and q respectively, if γ k is a geodesic joining p k and q k such that for every δ > k δ so that δ D ( γ k ( t )) ≤ δ for all k ≥ k δ , we can easily find an interval [ t k , s k ] such that π ( γ k ( t )) ∈ S for all t ∈ [ t k , s k ] and k γ ( t k ) − γ ( s k ) k ≥ c for some constant c > k large enough), getting a contradiction as before. (cid:3) In case the set of points of infinite type is not finite, the (bounded convex, smooth do-main) D might or might not have the visibility property. Roughly speaking, the visibilityproperty depends in a critical way on the rate of approach of the boundary to its complextangent space. We give an example below.Let ψ, χ , χ ∈ C ∞ ( R , R + ) be even functions, strictly increasing on R + . Let Ω ψ be aconvex domain such thatΩ ψ ∩ {k z k < } = (cid:8) z ∈ C : k z k < , Re z > ψ (Re z ) + χ (( | Im z | − + ) + χ (Im z ) (cid:9) . (5.4)We also assume that Ω ψ is smoothly bounded and contained in the ball B (0 , ∂ Ω ψ ∩ C × { } = [ − i, i ] × { } . We choose p := ( i, q := ( − i, χ , χ convex, we can ensure that ∂ Ω ψ is strictly pseudoconvex away from the line segment { } × [ − i ; 2 i ].If for some ǫ > Z ǫ ψ − ( u ) duu < ∞ , the domain Ω ψ satisfies the Goldilocks condition from Definition 2.6, and therefore hasthe visibility property, by [BZ, Theorem 1.4]. This is achieved, for instance, when ψ ( x ) ≥ exp (cid:0) − x (log x ) − α (cid:1) , for some α >
1, because then ψ − ( u ) ≤ (cid:0) log u (cid:1) (cid:0) log log u (cid:1) α . In contrast, we have
Proposition 5.4. If ψ ( x ) = o (cid:0) exp (cid:0) − π x (cid:1)(cid:1) near x = 0 , then the pair { p, q } does notverify the standard estimate, and therefore Ω ψ does not have the visibility property.Proof. Let p ε := ( i, ε ), q ε := ( − i, ε ), which tend to p and q respectively as ε →
0. Wehave,(5.5) k Ω ψ ( p ε , q ε ) ≤ log 2 + π ψ − ( ε ) . In order to prove (5.5), we will bound k Ω ψ ( p ε , q ε ) by constructing an almost explicitanalytic disc containing p ε and q ε . Let ω ε := { ζ ∈ C : ( ζ , ε ) ∈ Ω ψ } . For ε small enough,by convexity, ( C × { ε } ) ∩ Ω ψ ⊂ {k z k < } , so for those values of ε , ( ζ , ε ) ∈ Ω ψ if and onlyif ε > ψ ( | Re ζ | ) + χ (( | Im ζ | − + ) , and, in particular, ω ε ⊃ {| Im ζ | < , ψ (Re ζ ) < ε } =: R ε . ISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINS 19
Let ε ′ := ψ − ( ε ) and R ′ ε := iε ′ R ε = {| Re ζ | < ε ′ , | Im ζ | < } . Then k Ω ψ ( p ε , q ε ) ≤ k ω ε ( i, − i ) ≤ k R ε ( i, − i ) = k R ′ ε ( 1 ε ′ , − ε ′ ) . The conformal map φ ( z ) := e π z − e π z +1 maps the strip {| Im ζ | < } to the unit disc. Theline segments { Re ζ = ± ε ′ } are mapped to arcs of circles perpendicular to the unit circlewhich intersect the diameter ( − , +1) at the points e πε ′ − e πε ′ +1 and e − πε ′ − e − πε ′ +1 , so that φ ( R ′ ε ) ⊃ D (cid:0) , − e − πε ′ (cid:1) , while φ ( ± ε ′ ) = ± − e − π ε ′ e − π ε ′ .By renormalizing the smaller disc, for ε ′ small enough, k R ′ ε ( 1 ε ′ , − ε ′ ) ≤ k D (1 − e − π ε ′ , − e − π ε ′ ) = 2 k D (1 − e − π ε ′ , ≤ log 2 + π ε ′ , and (5.5) is proved.To conclude the proof, notice that12 log 1 δ Ω ψ ( p ε ) + 12 log 1 δ Ω ψ ( q ε ) = log 1 ε , and our hypothesis says that ε = o (cid:16) exp (cid:16) − π ψ − ( ε ) (cid:17)(cid:17) , i.e. log ε + π ψ − ( ε ) → −∞ as ε → −
12 log 1 δ Ω ψ ( p ε ) −
12 log 1 δ Ω ψ ( q ε ) + k Ω ( p ε , q ε ) ≤ log ε + π ψ − ( ε ) → −∞ , and we are done. (cid:3) Note that the previous condition is analogous to the “log-type” condition in [LW], butslightly less demanding. 6.
Localization
The results from Section 4 are obtained with the help of a global hypothesis, namelyconvexity of the whole domain. It seems natural to look for analogues with suitablylocalized hypotheses. We begin with a definition that ensures that a point p ∈ ∂D satisfies “half” of the standard estimate (2.3). Definition 6.1.
Let D be a domain in C n . We say that p ∈ ∂D is a weak k -point if forevery neighborhood W of p , lim sup z → p k D ( z, W c ) = ∞ . Moreover, p ∈ ∂D is a k-point if for every neighborhood W of p , lim inf z → p (2 k D ( z, W c ) + log δ D ( z )) > −∞ , where k D ( z, W c ) := inf w ∈ D \ W k D ( z, w ) . If both p, q are k -points, and p = q , then selecting disjoint neighborhoods W p and W q , the Kobayashi length of any curve connecting p ′ ∈ W p to q ′ ∈ W q is greater than k D ( p ′ , W cp ) + k D ( q ′ , W cq ) and so we see that the pair { p, q } satisfies the standard estimate.A weak k -point is a stronger kind of pseudoconvex point. In the spirit of the Konti-nuit¨atsatz, define (on a possibly non-smooth boundary) a non-pseudoconvex point as apoint such that there exists a family of analytic disks ( ϕ s ) s ∈ S such that ϕ s ( D ) ⊂ D , forany s ∈ S , ∪ s ∈ S ϕ s ( ∂ D ) is relatively compact in D , and p ∈ ∪ s ∈ S ϕ s ( D ).This definition makes it clear that a non-pseudoconvex point can not be a weak k -point, because the presence of discs inside D passing arbitrarily near p imply that pointsof D ∩ ∂W , for a small neighborhood W of p , stay at finite Kobayashi distance from pointsin D near p . In some cases, an obstruction is provided by the presence of an analytic discin ∂D . Lemma 6.2.
Let D ⊂ C n be a bounded domain. If ∂D is C -smooth, or if D is convex,and p lies in the interior of an affine analytic disc contained in ∂D , then p is not a weak k -point. This is proved in the case where ∂D is C by [Z2, Proposition 4.6].The converse is true for any C -convex domain, by [Z2, Proposition 3.5], which provesa stronger fact: if p is not a weak k -point, p must be in the interior of a convex face. Proof.
It is enough to “push” the analytic disc on ∂D inside the domain D by an arbitrarilysmall amount. Under each of the hypotheses, we will prove a slightly stronger propertyof the boundary:(PB) for any p ∈ ∂D , there exist a unit vector v ∈ C n , a neighborhood U of p , and apositive number ε such that : for any z ∈ U ∩ D , any t ∈ (0 , ε ), then z + tv ∈ D .If (PB) holds, let E be a complex affine disc centered at p , of radius r small enoughso that E ⊂ ∂D ∩ U . For t < ε , we have a family of complex affine discs parallel to E , E + tv . Take W := B ( p, r / r / p . Using the analytic discs E + tv we see that k D ( p + tv, ∂W ) remains bounded as ε → C boundary satisfies (PB) (this is almost stated in [JP, Remark3.2.3. (b)], and elementary). ISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINS 21
In the case where D is convex with no additional boundary smoothness, if p ∈ ∂D ,consider any p ∈ D . There is an r > B ( p , r ) ⊂ D .Consider F := (cid:0) ¯ B ( p , r/
2) + R + ( p − p ) (cid:1) ∩ ∂D . This is a closed neighborhood of p in ∂D . We can take for U a small enough open neighborhood of F , v := p − p k p − p k , and ε < dist( F, ¯ B ( p , r/ (cid:3) Corollary 6.3. If D is convex and bounded, p ∈ ∂D , then p is a weak k -point if and onlyif p is a peak point for A ( D ) . Indeed, by the above result, p is a weak k -point if and only if it lies in no non-trivialaffine analytic disc contained in ∂D , and by [Sib, Proof of Proposition 2.4] (see [NPZ,Proposition 10] for more details), this is equivalent to p being a holomorphic peak pointfor A ( D ) . Proposition 5.4 provides examples of points in the boundary of a convex domain, witha complex face reduced to a line segment, which are not k -points. On the other hand, ina convex boundary, having a trivial complex face is sufficient to obtain a k -point. Proposition 6.4. If D is convex and F p = { p } , then p is a k -point.Proof. The proof follows the lines of the proof of Theorem 4.1. We may restrict ourselvesto a relatively compact neighborhood W , and it will be enough to estimate k D ( z, w ) forany w ∈ D ∩ ∂W .For any H a supporting complex hyperplane for D , let π H be the projection parallel to H to a complex line orthogonal to H . For a point z ∈ D , choose a point p z ∈ ∂D suchthat k z − p z k = δ D ( z ). Then there is a unique supporting complex hyperplane H z at p z ,and k D ( z, w ) ≥ k H ( π H z ( z ) , π H z ( w )).Using the computation in (4.2), we will be done if we can prove that for a small neighbor-hood V of p , there exists η > z ∈ V , w ∈ ∂W , then dist( w, H z ) ≥ η . Sup-pose not: then we have sequences ( w k ) k ⊂ D ∩ ∂W and z k → p such that dist( w k , H z k ) →
0. Passing to subsequences, we may assume that ( H z k ) converges to a supporting complexhyperplane H ∞ through p and that w k → w ∞ ∈ H ∞ , therefore w ∞ ∈ ∂D ∩ ∂W . Thus w ∞ = p , and it is contained in the complex face H ∞ ∩ D , which is a contradiction. (cid:3) The next theorem and its proof are inspired by [LW, Theorem 1.4].
Theorem 6.5.
Let D be a domain in C n . Let U be a neighborhood of p ∈ ∂D such that p is a k -point for D U . Suppose either(1) that D U = D ∩ U is convex and bounded, or(2) that D U has α -growth for some α < and D is bounded.Then p is a k -point for D. Remark.
Any bounded convex domain has α ′ -log-growth for some α ′ > α -growth for any α >
0, so under both hypotheses above, D U has α -growth. The proof of Theorem 6.5 is deferred to Subsection 6.1.We would like to generalize this somewhat by using biholomorphisms.
Definition 6.6.
We say that p ∈ ∂D is a totally C -convexifiable point if there exists abiholomorphism Ψ from a bounded open neighborhood U of p to Ψ( U ) such that Ψ( D U ) =Ψ( D ∩ U ) is convex and F Ψ( p ) = { Ψ( p ) } , where the multiface is taken with respect to Ψ( D U ) . Note that any strictly pseudoconvex point is totally C -convexifiable. Theorem 6.7.
Let D be a domain in C n . Any totally C -convexifiable point p ∈ ∂D isa k -point for D . As a consequence, any pair of distinct totally C -convexifiable pointssatisfies the standard estimate. Since strictly pseudoconvex points are totally C -convexifiable, this provides a (modest)generalization of [FR, Corollary 2.4].The proof of Theorem 6.7 follows by adapting the proof of Theorem 6.5 and is given inSubsection 6.2.From Theorem 6.5 and the following localization formula (see [Lemma 2, Roy]):(6.1) κ D ( z ; X ) ≥ l D ( z, U c ) · κ D U ( z ; X ) , where l D ( z, U c ) = inf {| λ | : ∃ ϕ ∈ O ( D , D ) with ϕ (0) = z, ϕ ( λ ) ∈ D \ U } , we have Corollary 6.8.
Under the hypotheses of Theorem 6.5, there exists a neighborhood V ⊂⊂ U of p and a constant c > such that κ D ( z ; X ) ≥ (1 − cδ D ( z )) κ D U ( z ; X ) , z ∈ D V , X ∈ C n . The estimate in Corollary 6.8 is in the spirit of [FR, Theorem 2.1].Note that the hypotheses of Theorem 6.5 hold if D ∩ U is a bounded convex domain withvisibility property (or, more generally, if { p, q } has visible geodesics for any q ∈ ∂D ∩ U ).So, Corollary 6.8 substantially generalizes the localization given by [Theorem 3.2, LW].Assuming Dini-smooth regularity of ∂D near p, one may bootstrap the previous resultsto obtain a localization for k D which is inspired by [LW, Theorem 1.4]: the differencebetween the Kobayashi distances with respect to the local and global domains remainsbounded.It is easy to see that this localization is stronger than the one given by p being a k -pointfor D , but its proof uses Theorem 6.5, i.e. a weaker form of localization. Proposition 6.9.
Let D be a domain in C n with Dini-smooth boundary near p ∈ ∂D. Let U be a neighborhood of p such that p is a k -point for D U . Assume that D is boundedor that D U is convex and bounded. Then there exist a neighborhood W ⊂⊂ U of p and aconstant C > such that ≤ k D U ( z, w ) − k D ( z, w ) ≤ C, z, w ∈ D W . ISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINS 23
This is proved in Subsection 6.3.6.1.
The proof of Theorem 6.5.
Before starting the proof itself, we need two lemmasto compare the Kobayashi-Royden metrics of D and D U when z is near p , in the caseswhere D is bounded or when D U is convex. Lemma 6.10.
Suppose that D is bounded. Given any c < , there exist c ′ > and aneighborhood W of p , W ⊂⊂ U , such that for any z ∈ W , (6.2) κ D ( z ; X ) ≥ (1 − c ′ δ D ( z ) c ) κ D U ( z ; X ) , X ∈ C n . Proof.
Let V ⊂⊂ U. Since D is bounded, min z ∈ V l D ( z, U c ) =: c >
0. By (6.1), for z ∈ D V , (6.3) κ D ( z ; X ) ≥ c κ D U ( z ; X ) , X ∈ C n . Since k D is the integrated form of κ D ,k D ( z, V c ) ≥ c k D U ( z, V c ) , z ∈ D V . We may choose W ⊂⊂ V such that for z ∈ W , δ D U ( z ) = δ D ( z ). Since p ∈ ∂D U is a k -point, for z ∈ D W , we have l D ( z, U c ) ≥ l D ( z, V c ) ≥ tanh k D ( z, V c ) ≥ tanh c k D U ( z, V c ) ≥ − c ′ δ D ( z ) c , where c ′ depends on the constant implicit in Definition 6.1.Shrinking W further, we may reduce sup z ∈ W δ D ( z ) so that (6.3) holds for z ∈ W withany constant c < c . Repeating the previous argument oncemore, we get (6.2). (cid:3) Lemma 6.11.
Suppose that D U is convex. Given any c < , there exist c ′ > and aneighborhood W of p , W ⊂⊂ U , such that (6.2) holds for any z ∈ W .Proof. Since p is a k -point for the convex domain D U , by Corollary 6.3, p is a holomorphic,hence plurisubharmonic, peak point for D U . Then [Ber, Proposition 2.1.a] states that everysequence ( g k ) ∈ O ( D , D ) converges on compacta of D to p if and only if lim g k ( ζ ) = p forsome ζ ∈ D . Arguing by contradiction, we see as in [Ber, Proof of Proposition 2.1.b] thatfor any r ∈ (0 , U r of p such that for any map f ∈ O ( D , D ), f (0) ∈ D ∩ U r ⇒ f ( D (0 , r )) ⊂ D U . This implies that lim z → p l D ( z, U c ) = 1. So (6.1) showsthat for any c ∈ (0 ,
1) there exists a neighborhood V ⊂⊂ U of p such that (6.3) holds.Then we get the neighborhood W and the estimate (6.2) as in the proof of the previouslemma. (cid:3) Now we start the proof of Theorem 6.5.When D is bounded, we clearly have κ D ( z ; X ) ≥ c ′′ || X || . When D U is convex andbounded, by the above proof (6.3) holds for z ∈ D W , so, once again κ D ( z ; X ) ≥ c c ′′ || X || , z ∈ D W , X ∈ C n . We can then apply the following lemma, which is essentially proved in [BZ, Proposition4.4] (“existence of (1 , ε )-almost geodesic”).
Lemma 6.12.
Let D be a domain in C n and γ : [0 , → D be a C -curve with non-singularpoints. Assume that there exists a constant c > such that κ D ( γ ( t ); γ ′ ( t )) ≥ c || γ ′ ( t )) || forany t ∈ [0 , . Let σ : [0 , L ] → D , s σ ( s ) , be the parametrization of γ by Kobayashi-Royden length, ds = κ D ( γ ( t ); γ ′ ( t )) dt . Then σ is an absolutely continuous curve and κ D ( σ ( s ); σ ′ ( s )) = 1 for almost every s ∈ [0 , L ] . Let now z ∈ D W and w D W . By Lemma 6.12, for any ε > σ : [0 , L ] → D , with σ (0) = z and σ ( L ) = w , such that κ D ( σ ( s ); σ ′ ( s )) = 1 for almostevery s ∈ [0 , L ] and L ≤ k D ( z, w ) + ε .Set L ′ = inf { t ∈ (0 , L ] : σ ( t ) D W } and w ′ = σ ( L ′ ) . Choose t to maximize δ D ( σ ( t )) , t ∈ [0 , L ′ ] . Since D U has α -growth, it follows that(6.4) | t − t | − ε < k D ( σ ( t ) , σ ( t )) ≤ k D U ( σ ( t ) , z ) + k D U ( σ ( t ) , z ) ≤ βδ D ( σ ( t )) − α + βδ D ( σ ( t )) − α ≤ βδ D ( σ ( t )) − α . Define f ( t ) = 2 β if | t − t | < ε and f ( t ) = 2 β/ ( | t − t | − ε ) otherwise. Shrinking W as needed, we may assume that δ D ( σ ( t )) α ≤ f ( t ).Then using (6.2) k D U ( z, w ′ ) ≤ Z L ′ κ D U ( σ ( t ) , σ ′ ( t )) dt< Z L ′ (1 + 2 c ′ δ D ( σ ( t )) c ) κ D ( σ ( t ); σ ′ ( t )) dt = Z L ′ (1 + 2 c ′ δ D ( σ ( t )) c ) dt ≤ L ′ + 2 c ′ Z ∞ f ( t ) ν dt = k D ( z, W c ) + ε + C ε , where ν = c/α and C ε = 4 c ′ (2 β ) ν (cid:0) ε + ν − (cid:1) .Since ε > k D U ( z, W c ) ≤ k D ( z, W c ) + C , z ∈ D W which completes the proof of Theorem 6.5.6.2. The proofs of Theorem 6.7 and Theorem 1.4.
Proof of Theorem 6.7.
Here we show how to adapt the proof of Theorem 6.5 under thisgeneralized hypothesis.By Proposition 6.4, Ψ( p ) is a k -point for Ψ( D U ). As Ψ is a biholomorphism it onlychanges the distances to the boundary up to a fixed multiplicative constant near p , andthe Kobayashi distances are invariant, so p is a k -point for D U , and thus a peak point as ISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINS 25 well. Note that those properties are the only consequences of convexity used in the proofof Lemma 6.11.Furthermore, Ψ( D U ) has the α -growth property by convexity, and again Ψ being bi-holomorphism on U , this implies that D U has the α -growth property near p . This isexactly what is needed to complete the last part of the proof after Lemma 6.12.The second statement follows by the remark after Definition 6.1. (cid:3) Proof of Theorem 1.4.
Since every domain with Dini-smooth boundary has 1 / (cid:3) The proof of Proposition 6.9.
Since ∂D is Dini-smooth near p, by [NA, Theorem7], we may find a neighborhood W ⊂⊂ U of p such that k D U ( z, w ) ≤ log k z − w k p δ D ( z ) δ D ( w ) ! , z, w ∈ D W . On the other hand, by Theorem 6.5, p is a k -point for D . Then we may shrink W (ifnecessary) such that (6.2) and (6.4) hold for c = 1 and α < , respectively, and thefollowing is true: if ε ∈ (0 ,
1) and γ : [0 ,
1] is a piecewise C -curve with γ (0) = z ∈ D W ,γ ( L ) = w ∈ D W , and R L κ ( γ ( t ); γ ′ ( t )) dt < k D ( γ (0) , γ (1)) + ε, then γ ([0 , L ]) ⊂ U. Thenfor σ as in Lemma 6.12 (with L ′ = L and w ′ = w ), References [Aba] M. Abate,
Iteration theory of holomorphic maps on taut manifolds , Research and Lecture Notes inMathematics. Complex Analysis and Geometry, Mediterranean Press, Rende, 1989.[AR] M. Abate, J. Raissy,
Wolff-Denjoy theorems in nonsmooth convex domains . Ann. Mat. Pura Appl.193 (2014), 1503–1518.[BB] Z. M. Balogh, M. Bonk,
Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvexdomains , Comment. Math. Helv. 75 (2000), 504–533.[Ber] F. Berteloot,
Characterization of models in C by their automorphisms groups , Internat. J. Math.5 (1994), 619–634.[BM] G. Bharali, A. Maitra, A weak notion of visibility, a family of examples, and Wolff-Denjoy theorems ,Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), https://doi.org/10.2422/2036-2145.201906 007.[BZ] G. Bharali, A. Zimmer,
Goldilocks domains, a weak notion of visibility, and applications , Adv.Math. 310 (2017), 377–425.[Blo] Z. B locki,
The Bergman metric and the pluricomplex Green function , Trans. Amer. Math. Soc. 357(2005), 2613–2625.[BCD] F. Bracci, M. D. Contreras, S. Diaz-Madrigal,
Continuos Semigroups of holomorphic self-maps ofthe unit disc , Springer Monographs in Mathematics, Springer Nature Switzerland AG, 2020.[BF] F. Bracci, J. E. Fornæss,
The range of holomorphic maps at boundary points , Math. Ann. 359(2014), 909–927.[BFW] F. Bracci, J. E. Fornæss, E. F. Wold,
Comparison of invariant metrics and distances on stronglypseudoconvex domains and worm domains , Math. Z. 292 (2019), 879–893 (2019).[BG] F. Bracci, H. Gaussier,
Horosphere Topology , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 20 (2020),239–289. [BGZ] F. Bracci, H. Gaussier, A. Zimmer,
Homeomorphic extension of quasi-isometries for convex do-mains in C d and iteration theory , Math. Ann. (2020), https://doi.org/10.1007/s00208-020-01954-1.[BST] F. Bracci, A. Saracco, S. Trapani, The pluricomplex Poisson kernel for strongly pseudoconvexdomains , arXiv:2007.06270, to appear in Adv. Math.[DFW] K. Diederich, J. E. Fornæss, E. F. Wold,
Exposing points on the boundary of a strictly pseudo-convex or a locally convexifiable domain of finite 1-type . J. Geom. Anal. 24 (2014), 2124–2134.[Fia] M. Fiacchi,
Gromov hyperbolicity of pseudoconvex finite type domains in C , Math. Ann. (2021),https://doi.org/10.1007/s00208-020-02135-w.[FR] F. Forstneric, J.-P. Rosay, Localization ot the Kobayashi metric and the boundary continuity ofproper holomorphic mappings , Math. Ann. 279 (1987), 239–252.[Gau] H. Gaussier,
Characterization of convex domains with non compact automorphism group , Mich.Math. J. 44 (1997), 375–388.[JP] M. Jarnicki, P. Pflug,
Invariant distances and metrics in complex analysis – 2nd extended edition ,de Gruyter, 2013.[LW] J. Liu, H. Wang,
Localization of the Kobayashi metric and applications , Math. Z. (2020),https://doi.org/10.1007/s00209-020-02538-0.[Nik] N. Nikolov,
Estimates of invariant metrics on “convex domains” , Ann. Mat. Pura Appl. 193 (2014),1595–1605.[NA] N. Nikolov, L. Andreev,
Estimates of the Kobayashi and quasi-hyperbolic distances , Ann. Mat. PuraAppl. 196 (2017), 43–50.[NPT] N. Nikolov, P. Pflug, P.J. Thomas,
Upper bound for the Lempert function of smooth domains ,Math. Z. 266 (2010), 425–430.[NPZ] N. Nikolov, P. Pflug, W. Zwonek,
Estimates for invariant metrics on C -convex domains , Trans.Amer. Math. Soc. 363 (2011), 6245–6256.[NT] N. Nikolov, M. Trybu la, The Kobayashi balls of ( C -)convex domains ,Monatsh. Math. 177 (2015),627–635.[Roy] H. Royden, Remarks on the Kobayashi metric , in “Several Complex Variables II”, Lecture Notesin Math. 185 (1971), Springer Verlag, Berlin, 125–137.[Sib] N. Sibony,
Une classe de domaines pseudoconvexes , Duke Math. J. 55 (1987), 299–319.[Ven] S. Venturini,
Pseudodistances and pseudometrics on real and complex manifolds , Ann. Math. PuraAppl. 154 (1989), 385–402.[Z1] A.M. Zimmer,
Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type ,Math. Ann. 365 (2016), 1425–1498.[Z2] A.M. Zimmer,
Gromov hyperbolicity, the Kobayashi metric, and C -convex sets , Trans. Amer. Math.Soc. 369 (2017), 8437–8456.[Z3] A.M. Zimmer, Characterizing domains by the limit set of their automorphism group , Adv. Math.308 (2017), 438–482.
F. Bracci: Dipartimento Di Matematica, Universit`a di Roma “Tor Vergata”, Via DellaRicerca Scientifica 1, 00133, Roma, Italy
Email address : [email protected] N. Nikolov: Institute of Mathematics and Informatics, Bulgarian Academy of Sci-ences, Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria,Faculty of Information Sciences, State University of Library Studies and InformationTechnologies, 69A, Shipchenski prohod Str., 1574 Sofia, Bulgaria
Email address : [email protected] ISIBILITY OF KOBAYASHI GEODESICS IN CONVEX DOMAINS 27
P.J. Thomas: Institut de Math´ematiques de Toulouse; UMR5219, Universit´e de Toulouse;CNRS, UPS, F-31062 Toulouse Cedex 9, France
Email address ::