A Kobayashi metric version of Bun Wongs theorem

A K OBA Y ASHI METRIC VE RSION OF BUN W ONG’S THEOREM KANG-T AE KIM AND STEVEN G. KRANTZ 1. Intr oduction 1.1. B asic T ermin ology and Statemen t of Main Theorem. F or a complex manifold M , denote by k M its Kobay ashi-Ro yden infinitesimal metric and b y d M its Kobay ashi distance, and (see [KRA1], [KOB2]). Definition 1.1. A map f : M → N from a comple x manifold M in to another complex manifold N is said to b e a Kob ayashi isometry if f is a homeomorphism satisfying the conditio n that d N ( f ( x ) , f ( y )) = d M ( x, y ) for ev ery x, y ∈ M . The set G M of Ko bay ashi isometries o f M (onto M itself ) endow ed with the compact-op en top ology is a top ological group with resp ect to the binary law of comp osition of mappings. W e call t his group G M , the Kob ayashi isometry gr oup of the complex manif o ld M . Denote b y B n the op en unit ball in C n . Notice that the K o ba yashi distance of B n in fact coincides with t he P oincar´ e-Bergman distance of B n . The primary aim of this article is to establish the fo llo wing theorem, whic h is a Koba y a shi metric v ersion of Bun W ong’s classical theorem [W O N ]. Theorem 1.1. L et Ω b e a b ounde d dom ain in C n with a C 2 ,ǫ smo oth ( ǫ > 0 ), str ongly pseudo c onvex b oundary. If its Kob ayashi iso m etry gr oup G Ω is non-c om p act, then Ω is bi h olomorphic to the op e n unit b al l B n . The resear ch of the first named author is supp orted in part b y the KOSEF Gra n t R01-20 05-00 0 -10771-0 of The Republic of Ko rea. He a ls o would like to thank the Cent re de Mathematiques et Informatiques, Universit ´ e de Provence, Aix-Mars e ille 1 (F rance ) for its hospitality at the final stage of this work. Both authors thank the American Institute o f Mathematics for its hospitality during a p ortion of this work. The second named autho r ha s grants from the Natio nal Science F o undation and the Dean of the Gra dua te Scho ol at W ashing to n University . 1 2 KANG-T AE KIM AND STEVEN G. KRANTZ Remark 1.1. The classical results (see [WON] and [R OS]) a ssume noncompact (biholomorphic) automorphism group of the domain Ω. But it has b een understo o d fo r man y y ears that this theorem of Bun W ong and Rosa y is really a “flattening” result of geometry (terminology of Gr omo v is b eing used here). Our new theorem puts this relationship in to p erspectiv e.  1.2. The Koba y ashi-distance v ersion of W ong’s theorem. Ex- p erts who a re familiar with W ong’s theorem [W ON] w ould exp ect the follo wing: Theorem 1.2 (Seshadri-V erma) . L e t Ω b e a b ounde d domai n in C n with a C 2 smo oth, str ongly pseudo c onvex b oundary. If its Kob ayashi isometry gr oup G Ω is no n -c omp act, then ther e exists a Kob ayashi isom- etry f : Ω → B n . Certainly it was Seshadri and V erma ([SEV1, SEV2] as w ell as [VER]) who first conceiv ed the idea o f a metric vers ion of W ong’s the- orem. W e prese n t here a different pro of of T heorem 1.2 which closes some gaps in the extant ar g umen t a nd answe rs some subtle questions. W e b eliev e that the argumen ts w e presen t in this art icle can b e of use for many other purp oses. The a rgumen ts in v olved here are subtle, b e- cause the mappings under consideration are a priori only contin uous. Unlik e the holomorphic case the restrictions of Ko bay ashi isometries to sub-domains are not isometries with resp ect to t he Koba y ashi metric of the sub-domain. Another slipp ery p oint is that the full p o we r of Mon tel’s theorem and Cartan’s uniqueness theorem is not av a ilable for equi-con tinuous maps. So it is necessary to give precise estimates and argumen t s that clarify all these subtle p oin ts necessary for the pro of. It is true that our pro of of this theorem fo llo ws the same general line of reasoning as [SEV1, SEV2], whic h in turn is along the line of scaling metho d in tro duced by S. Pinc huk around 1980 ([PIN]). The second ha lf, that is indeed the main part of this article, presen ts the follow ing: Theorem 1.3. L et Ω b e a b ounde d domain in C n with a C 2 ,ǫ smo oth, str ongly pse udo c onvex b oundary. If ther e exists a Kob ayashi isometry f : Ω → B n fr om Ω onto the op en unit b al l B n of C n , then f is either holomorphic or c onjugate holomorphi c . Notice that Theorems 1.2 and 1 .3 imply Theorem 1 .1. And that is the main p oin t of this pap er. The w ork [SEV2], whic h addresses similar questions, assumes that the mapping is C 1 to the b o undary . W e are able to eliminate this somewhat indelicate h yp o thesis. A KOBA Y AS HI METRIC V ERSION OF BUN WONG’S THEOREM 3 Seshadri and V erma in the ab ov e cited w or k hav e prov ed the same conclusion in the case when Ω is strongly conv ex. Notice that o ur theorem here assumes only str ong pseudo c onvexity . On t he other hand, the exp erts in this line of researc h w ould feel that the o ptimal regula rit y of the b o undar y should b e C 2 , instead of C 2 ,ǫ with some ǫ > 0. A t the time of this writing, w e do not kno w how to achie v e t he optimum , b ecause of tech nical reasons connected with w o r k of Lemp ert [LEM1, LEM2]. W e would like to men tion it as a question for future study . 2. Some Fund ament als 2.1. T erminology and Notation. Let Ω b e a Kobay ashi hy p erb olic domain in C n . F or a p oint q ∈ Ω, let us write G Ω ( q ) = { ϕ ( q ) | ϕ ∈ G Ω } . This is usually called the (p oint) orbit o f q under the action of the Koba y a shi isometry group G Ω on the domain Ω. Call a b oundar y p oin t p ∈ ∂ Ω a b ounda ry o rbit ac cumulation p oint if p b elongs to t he closure G Ω ( q ) of the orbit G Ω ( q ) of a certain in terior p oin t q ∈ Ω under the action of the K oba y ashi isometry group G Ω . In other words, p is a b oundary orbit accumulation p oin t if and only if there exists an in terior p oint q ∈ Ω and a sequence of Kobay ashi isometries ϕ j ∈ G Ω suc h that lim j →∞ ϕ j ( q ) = p . Let us adopt the notation B d ( q ; r ) := { y | d ( y , q ) < r } for an y distance d in general. Then one observ es the f o llo wing: Prop osition 2.1. L et Ω b e a b ounde d, c omplete Kob ayash i hyp erb ol i c domain in C n . Then i ts Kob ayash i isom etry gr oup G Ω is no n-c omp act if an d only if Ω admits a b oundary orbi t a c cumulation p o int. Pr o of. Notice that the sufficiency is obv ious. W e establish the ne c essity only . Exp ecting a contradiction, assume to the contrary that there a r e no b oundary o rbit accum ulatio n p oints. Then, f or eve ry p oint q of the domain Ω, the or bit o f q under the gr o up a ctio n is relatively compact. No w let { ϕ j } b e an arbitrarily c hosen sequence of Kobay ashi isome- tries; t hen it is ob viously an equi-con tinuous family with resp ect to the Koba y a shi distance. By Barth’s theorem ([BAR]), this implies that ϕ j forms an equi-con t inuous family on compact subsets with resp ect to the Euclidean distance. Th us o ne ma y use the Arzela-Ascoli theorem 4 KANG-T AE KIM AND STEVEN G. KRANTZ to extract a sequence { ϕ j k } that con v erges uniformly on compact sub- sets of Ω to a limit mapping b ϕ . Th us, r eplacing ϕ j b y a subsequence, one ma y assume without loss of generalit y that ϕ j con verges uniformly on compacta to a contin uous map, say b ϕ . Since the p o in t orbit is alw a ys compact, b ϕ ( a ) = b for some a, b ∈ Ω. Notice that, exploiting the completeness of d Ω , one can deduce that b ϕ (Ω) = b ϕ  ∞ [ ν =1 B d Ω ( a ; ν )  = ∞ [ ν =1 b ϕ ( B d Ω ( a ; ν )) ⊂ ∞ [ ν =1 B d Ω ( b ; ν ) = Ω . It is obvious tha t one can apply the same a r g umen t to the sequence ϕ − 1 j (replacing it b y a subsequence that con verges uniformly on compacta, if necessary). Th us b ϕ : Ω → Ω is a ho meomorphism. It is ob vious that b ϕ preserv es the Koba y ashi distance d Ω . Altogether, it follo ws that b ϕ ∈ G Ω . This establishes that G Ω is compact. (This argumen t sho ws the sequen tial compactness of G Ω , t o b e precise. But then G Ω equipped with the top olog y of unifor m conv ergence on compacta is metrizable.) This con tradiction yields the desired conclusion.  Since eve ry b ounded strongly pseudo conv ex domain is complete with resp ect to the Koba y ashi distance ( see [GRA]), Theorem 1.2 no w follo ws b y the follow ing more general statemen t: Theorem 2.1. L et Ω b e a domain in C n . I f Ω admits a b oundary orbit ac cumulation p oint at w h ich the b oundary of Ω is C 2 smo oth, str ongly pseudo c onvex, then Ω is Kob ayashi isometric to the op e n unit b al l in C n . Notice that for this theorem one do es not need to assume that the domain has t o b e a prio ri b ounded or Kobay ashi h yp erb olic. Instead, the complete Kobay ashi h yp erb o licity o f the do main will b e obtained along the w a y in the pro of , f r om t he giv en hypothesis only . The pro of of this result will b e dev elop ed in Section 3. 2.2. Some Comparison Estimates. W e no w give some comparison inequalities f or the K o ba y ashi distances and the Koba y ashi-R o yden infinitesimal metrics fo r a sub-domain against its a m bient domain. This will pla y a crucial role in the pro of of Theorem 2.1. Lemma 2.1. (Kim-Ma) L et Ω b e a Kob ayashi hyp erb olic dom ain in C n with a sub d o main Ω ′ ⊂ Ω . L et q , x ∈ Ω ′ , let d Ω ( q , x ) = a , and let b > a . If Ω ′ satisfies the c ondition B d Ω ( q ; b ) ⊂ Ω ′ , then the fol lowing A KOBA Y AS HI METRIC V ERSION OF BUN WONG’S THEOREM 5 two ine qualities hold: d Ω ′ ( q , x ) ≤ 1 tanh( b − a ) d Ω ( q , x ) , k Ω ′ ( x, v ) ≤ 1 tanh( b − a ) k Ω ( x, v ) , v ∈ C n . [R e c al l her e that k is the infinitesima l Kob ayashi/R o yden metric and d is the inte gr ate d Kob ayashi/R oyden distanc e.] Pr o of. F or the sak e of the reader’s conv enience, we include here the pro of, lifting it from [KIMA]. Let s = tanh( b − a ) a nd let ǫ > 0. Denote b y ∆ the o p en unit disc in C and by ∆( a ; r ) the op en disc of Euclidean radius r cen tered at a in C . Then, b y definition of k Ω ( x, v ), there exists a holomorphic map h : ∆ → Ω suc h that h (0) = x and h ′ (0) = v / ( k Ω ( x, v ) + ǫ ). If ζ ∈ ∆(0; s ), then d Ω ( q , h ( ζ )) ≤ d Ω ( q , x ) + d Ω ( x, h ( ζ )) = a + d Ω ( h (0) , h ( ζ )) ≤ a + d ∆ (0 , ζ ) < a + ( b − a ) = b. This sho ws that h ( ∆(0 ; s )) ⊂ Ω ′ . No w define g : ∆ → Ω ′ b y g ( z ) := h ( sz ). Then one has g (0) = x a nd g ′ (0) = sh ′ (0) = sv / ( k Ω ( x, v ) + ǫ ). This implies that k Ω ′ ( x, v ) ≤ ( k Ω ( x, v ) + ǫ ) /s . Since ǫ is a n arbitrarily c hosen p ositiv e num b er, it follows that k Ω ′ ( x, v ) ≤ 1 tanh( b − a ) k Ω ( x, v ) , ∀ v ∈ C n . No w let δ b e chose n suc h tha t 0 < δ < b − a . There is a C 1 curv e γ : [0 , 1] → Ω suc h that γ (0) = q , γ (1 ) = x , and Z 1 0 k Ω ( γ ( t ) , γ ′ ( t )) dt < a + δ . This implies that d Ω ( q , γ ( t )) < a + δ < b for any t ∈ [0 , 1]. Hence γ ( t ) ∈ Ω ′ for ev ery t ∈ [0 , 1]. Notice that the inequalit y k Ω ′ ( γ ( t ) , γ ′ ( t )) ≤ k Ω ( γ ( t ) , γ ′ ( t )) / tanh( b − a − δ ) holds for ev ery t ∈ [0 , 1] b y the preceding a rgumen ts. But then t his implies that d Ω ′ ( x, q ) ≤ Z 1 0 k Ω ′ ( γ ( t ) , γ ′ ( t )) dt ≤ 1 tanh( b − a − δ ) Z 1 0 k Ω ( γ ( t ) , γ ′ ( t ) dt. 6 KANG-T AE KIM AND STEVEN G. KRANTZ Consequen tly , one deduce s that d Ω ′ ( x, q ) < ( a + δ ) / tanh( b − a − δ ). No w, letting δ t end t o 0, one obtains the desired conclusion.  3. Scaling With a Sequence of Koba y ashi Isometries; Proo f of Theorem 2.1 No w w e presen t a precise and detailed pro of of Theorem 2.1. Denote b y B ( p ; r ) = { z ∈ C n | | z − p | < r } , the op en ball o f radius r cen tered at p with resp ect to the Euclidean distance on C n . Because of the C 2 strong pseud o conv exit y of ∂ Ω at the b o undary orbit accum ulation p oin t p , there exist a p ositiv e real nu m b er ε and a biholomorphic mapping Ψ : U → B (0; ε ) suc h that Ψ( p ) = 0, Ψ(Ω ∩ U ) = { ( z 1 , . . . , z n ) ∈ B (0; ε ) | Re z 1 > | z 1 | 2 + . . . + | z n | 2 + E ( z 1 , . . . , z n ) } , and Ψ( ∂ Ω ∩ U ) = { ( z 1 , . . . , z n ) ∈ B (0; ε ) | Re z 1 = | z 1 | 2 + . . . + | z n | 2 + E ( z 1 , . . . , z n ) } , where E ( z 1 , . . . , z n ) = o ( | z 1 | 2 + . . . + | z n | 2 ) . Apply now the lo calization metho d b y N. Sib o ny that uses o nly the plurisubharmonic p eak functions. (See [SIB], [BER], [GA U], [BYGK] for instance, a s w ell as [R OY].) It follo ws from the hypothesis that ev ery op en neigh b orho o d U of p in C n admits an op en set V in C n suc h that p ∈ V ⊂⊂ U and 1 2 k Ω ∩ U ( z , ξ ) ≤ k Ω ( z , ξ ) for eve ry z ∈ Ω ∩ V and ev ery ξ in the tangen t space T z Ω (= C n ) of Ω at the p oint z ∈ Ω. It can also b e arra ng ed that 1 2 d Ω ∩ U ( x, y ) ≤ d Ω ( x, y ) for ev ery x, y ∈ Ω ∩ V . See [BYGK] for instance for this last inequalit y . This in particular implies A KOBA Y AS HI METRIC V ERSION OF BUN WONG’S THEOREM 7 The lo calization prop ert y : F or an y op en neigh b orho o d W of p and for an y relat iv ely c ompact s ubset K o f Ω , there exists a p ositiv e in teger j 0 suc h that ϕ j ( K ) ⊂ Ω ∩ W whenev er j > j 0 . Notice that one can tak e W suc h that Ω ∩ W equipped with the Koba y a shi distance d Ω ∩ W is Cauc h y complete. Conseq uen tly the lo- calization prop ert y , together with the fact that p is a b o undar y or bit accum ulation p oint, implies that Ω is complete Ko ba yashi h yp erb olic. Next w e apply Pinc huk’s scaling metho d [PIN]. Let Ψ ◦ ϕ j ( q ) ≡ ( q 1 j , . . . , q nj ) for j = 1 , 2 , . . . . F ix j for a moment. Cho ose p 1 j ∈ C suc h that ( p 1 j , q 2 j , . . . , q nj ) ∈ Ψ( ∂ Ω ∩ U ) and q 1 j − p 1 j > 0 . Let ζ = A j ( z ) for the complex a ffine map A j : C n → C n defined b y ζ 1 = e iθ j ( z 1 − p 1 j ) + n X k =2 c k j ( z k − q k j ) ζ 2 = z 2 − q 2 j . . . ζ n = z n − q nj , where the real num b er θ j and the complex n umbers c 2 j , . . . , c nj are c hosen so tha t the real h yp ersurface A j ◦ Ψ( ∂ Ω ∩ U ) is tangent to the real h yp erplane defined b y the equation R e ζ 1 = 0. It is imp ortan t to notice no w, for the computation in the later part of this pro of, tha t lim j →∞ e iθ j = 1 and lim j →∞ c mj = 0 for ev ery m ∈ { 2 , . . . , n } . Then define Λ j : C n → C n b y Λ j ( z 1 , . . . , z n ) = z 1 λ j , z 2 p λ j , . . . , z n p λ j ! , where λ j = q 1 j − p 1 j . 8 KANG-T AE KIM AND STEVEN G. KRANTZ Exploit no w the m ulti-v ariable Cayley tr a nsform Φ( z 1 , . . . , z n ) =  1 − z 1 1 + z 1 , 2 z 2 1 + z 1 , . . . , 2 z n 1 + z 1  , and consider the followin g sequences σ j := Φ ◦ Λ j ◦ A j ◦ Ψ | U , and τ j := σ j ◦ ϕ j , for j = 1 , 2 , . . . . Notice that eac h σ j maps U in to C n . It plays the role of a holo- morphic embedding of Ω ∩ U in to C n . On the other hand, the domain of definition of τ j has to b e considered more carefully . T hanks to the lo c alization pr op erty ab ov e, f o r ev ery compact subset K of Ω, there exists a p ositive in teger j ( K , U ) suc h that τ j maps K in to C n for ev- ery j ≥ j ( K , U ). Th us, for eac h suc h K , one is allo we d to consider τ j | K : K → C n only for the indic es j with j ≥ j ( K , U ). No w a direct c alculation sho ws that, shrinking U if nec essary , f or ev ery ǫ > 0 there exists a p ositiv e integer N suc h that B (0; 1 − ǫ ) ⊂ σ j (Ω ∩ W ) ⊂ B (0; 1 + ǫ ) for ev ery j > N . Th us, replacing N by N + j ( K , U ), one may conclude that τ j ( K ) ⊂ B (0; 1 + ǫ ) for ev ery j > N . T ak e no w a sequence { K ν } of relativ ely compact subsets o f Ω satis- fying the follo wing three conditions: (i) K ν is a relativ ely compact, op en subset of Ω for eac h ν ; (ii) K ν ⊂ K ν +1 , for ν = 1 , 2 , . . . ; (iii) ∞ [ ν =1 K ν = Ω. Suc h a sequen ce { K ν } is usually called a (r elatively) c omp act exhaustion se quenc e o f Ω. Giv en a relatively compact exhaustion sequence { K ν } of Ω, we con- sider the restricted sequences { τ j,ν = τ j | K ν | j = 1 , 2 , . . . } , for ev ery ν = 1 , 2 , . . . . Using these restricted sequences, we would lik e to establish: A KOBA Y AS HI METRIC V ERSION OF BUN WONG’S THEOREM 9 Claim ( † ): There exists a compact exh austion sequence { K ν } suc h that the sequence τ j admits a subse quence that con v erges unifor mly on ev ery compact subset o f Ω to a Ko ba y ashi isometry b τ : Ω → B from the domain Ω on to the ball B . Notice that the pro of of Theorem 2.1 is complete as so on as this Claim is established. Apply no w Lemma 2.1 to the domain Ω. Let ν b e an a rbitrarily c hosen p ositiv e in teger. Le t z 0 ∈ Ω. Let the K o ba y ashi metric ball B d Ω ( z 0 ; µν ) play the role of t he sub do main Ω ′ , where µ is an integer with µ > 5. Then, for an y x, y ∈ B d Ω ( z 0 , ν ), it holds that d B d Ω ( z ; 2 µν ) ( x, y ) ≤ 1 tanh( µν ) d Ω ( x, y ) . Exploiting the fact that (Ω , d Ω ) is Cauc h y complete, w e now c ho ose the relativ ely compact exhaustion sequence consisting o f expanding Koba y a shi metric balls: K ν ≡ B d Ω ( q ; ν ) . T ak e N > 0 suc h that ϕ j ( q ) ∈ V ∩ Ω and ϕ j ( K 2 µν ) ⊂ Ω ∩ U whenev er j > N . Enlarging N if necessary , we may ac hiev e also that σ j (Ω ∩ U ) ⊂ B (0; 1 + ǫ ) fo r eve ry j > N . Moreo ve r, for an y x, y ∈ K ν , one sees that: d B (0;1+ ǫ ) ( τ j ( x ) , τ j ( y )) ≤ d σ j (Ω ∩ U ) ( σ j ◦ ϕ j ( x ) , σ j ◦ ϕ j ( y )) = d Ω ∩ U ( ϕ j ( x ) , ϕ j ( y )) ≤ d B d Ω ( ϕ j ( q ) , ; 2 µν )) ( ϕ j ( x ) , ϕ j ( y )) ≤ 1 tanh( µν ) d Ω ( ϕ j ( x ) , ϕ j ( y )) = 1 tanh( µν ) d Ω ( x, y ) . As a summary , w e hav e that (1) d B (0 , 1+ ǫ ) ( τ j ( x ) , τ j ( y )) ≤ 1 tanh( µν ) d Ω ( x, y ) , ∀ x, y ∈ K ν . This es timate show s that the seq uence { τ j } is a n eq ui-con tin uous family on eac h K ν . Therefore one ma y extract a subsequence that 10 KANG-T AE KIM AND STEVEN G. KRANTZ con verges uniformly on ev ery compact subset of Ω to a contin uous map b τ : Ω → B (0; 1 + ǫ ). No w the ab ov e estimate yields (2) d B (0;1+ ǫ ) ( b τ ( x ) , b τ ( y )) ≤ 1 tanh( µν ) d Ω ( x, y ) , ∀ x, y ∈ K ν . Since this e stimate m ust hold for ev ery ǫ > 0, one deduces first that b τ ( K ν ) ⊂ B (0; 1) = B . But then, using the distance estimate ab ov e one sees immediately that τ j ( K ν ) for any j is b ounded aw a y from the b oundary of B . So b τ ( K ν ) ⊂ B for ev ery ν . Conse quen tly b τ ma ps Ω in to B . Moreo ver, d B ( b τ ( x ) , b τ ( y )) ≤ 1 tanh( µν ) d Ω ( x, y )) , ∀ x, y ∈ K ν . Letting µ → ∞ , this last estimate turns in t o (3) d B ( b τ ( x ) , b τ ( y )) ≤ d Ω ( x, y )) , ∀ x, y ∈ K ν . No w let x, y ∈ Ω b e fixed. Ch o ose 0 < δ < 1 / 2 suc h that b τ ( x ) , b τ ( y ) ∈ B (0; 1 − 2 δ ). Then there exists a p ositiv e integer N 1 suc h that τ j ( x ) , τ j ( y ) ∈ B (0; 1 − δ ) and B (0; 1 − δ ) ⊂ σ j (Ω ∩ U ) whenev er j > N 1 . Th us o ne obtains d Ω ( x, y ) = d Ω ( ϕ j ( x ) , ϕ j ( y )) ≤ d Ω ∩ U ( ϕ j ( x ) , ϕ j ( y )) ≤ d σ j (Ω ∩ U ) ( σ j ◦ ϕ j ( x ) , σ j ◦ ϕ j ( y )) = d σ j (Ω ∩ U ) ( τ j ( x ) , τ j ( y )) ≤ d B (0;1 − δ ) ( τ j ( x ) , τ j ( y )) . Let j tend to infinity fir st, and then let δ con v erge to zero. Then one deduces that (4) d Ω ( x, y ) ≤ d B ( b τ ( x ) , b τ ( y )) . Com bining (3) and (4), o ne sees that d Ω ( x, y ) = d B ( b τ ( x ) , b τ ( y )) . Since x and y hav e b een arbitr arily chos en p oin ts of Ω, it follo ws that b τ : Ω → B preserv es the Kobay ashi distance. A KOBA Y AS HI METRIC V ERSION OF BUN WONG’S THEOREM 11 In order to complete the pro of of the claim a nd hence Theorem 2.1, it remains to sho w that b τ : Ω → B is surjectiv e. Let y ∈ B . Then there exists r with 0 < r < 1 suc h t hat | y | < r . Moreov er, there exists N 2 > 0 suc h that τ − 1 j ( y ) ∈ Ω fo r eve ry j > N 2 . Let x j = τ − 1 j ( y ). Then it ho lds that d Ω ( q , x j ) ≤ d B (0; y ) + 1 for ev ery j > N 2 . Therefore a subsequence x j k of x j con verges , sa y , to b x ∈ Ω. No w, b ecause of the unifo r m con vergenc e of τ j to b τ on compacta, one immediately sees that b τ ( b x ) = b τ ( lim j →∞ ( x j )) = [ lim k →∞ τ k ]( lim j →∞ ( x j )) = lim j →∞ τ j ( x j ) = y . This shows that b τ : Ω → B is surjectiv e. Consequen tly , the proo f of Claim ( † ) follows . The pro of of Theorem 2.1 is now complete.  4. Complex Anal yticity of t he K oba y ashi Isometr y f : Ω → B n It no w remains to establish Theorem 1.3. Inciden tally , it seems appropriate for us to p o se the follo wing naturally arising question: Question 4.1. Let n b e a p ositiv e integer. Let Ω 1 and Ω 2 b e b ounded domains in C n with C 2 smo oth, strongly pseudo conv ex b oundaries, and let f : Ω 1 → Ω 2 b e a homeomorphism that is an isometry with resp ect to the Kobay ashi distances. Then, is f or f necessarily holomorphic? W e do not kno w the answ er to this question a t presen t; w e show in this pap er that the answ er is affirmat iv e in case Ω 2 = B n and ∂ Ω 1 is C 2 ,ǫ smo oth. 4.1. B urns-Kran tz construction of Lemp ert discs for strongly pseudoc onv ex domains. Here w e would lik e to explain ho w Burns and Krantz adapted Lemp ert’s analysis to the strong ly pseudo con v ex domains, as this set of ideas is go ing to play an imp ortant ro le fo r our pro of. In what follo ws, let Ω ⊆ C n b e a b ounded, strongly pseudo con- v ex domain with C 2 ,ǫ b oundary . Let p ∈ ∂ Ω. Then by Burns and Krantz [BUK], there ex ist open neigh b orho o ds V and U of p suc h that p ∈ V ⊂⊂ U suc h that an y p ′ ∈ ∂ Ω ∩ V admits a Lemp ert disc ϕ : ∆ → Ω suc h that p ′ ∈ ϕ ( ∂ ∆) and ϕ ( ∆) ⊂ U . Perhaps the term Lemp ert disc needs to b e clarified. In fa ct w e will do more than that. W e will quic kly describ e what Burns and Kran tz presen t in Prop osition 4.3 and Lemma 4.4 of [BUK]. T ak e the F ornaess em b edding ([FOR]) that em b eds Ω holomor phi- cally and prop erly into a strongly conv ex domain Ω ′ ⊂ C N with some N > n . This em b edding map, sa y F , is in fact smo oth up to the 12 KANG-T AE KIM AND STEVEN G. KRANTZ b oundary of Ω suc h that F : Ω → Ω ′ is smo oth, and also F ( ∂ Ω) ⊂ ∂ Ω ′ . Let T F ( p ) ( F (Ω)) be t he tangent pla ne to F (Ω) at F ( p ). This is an n -dimensional complex affine space in C N and so w e ma y iden tify it with the standard C n . Let Π : C N → T F ( p ) ( F (Ω)) b e the orthogonal pro jection. Then Π ◦ F : Ω → C n is a ho lo morphic mapping, and fur- thermore is a injectiv e holomorphic mapping of Ω ∩ U for some op en neigh b orho o d U of p in C n . Denote b y F ′ := Π ◦ F . Since F ′ (Ω) is b ounded, there exists a sufficien tly la rge ball B suc h that Ω ⊂ B and p ∈ ∂ Ω ∩ ∂ B . Slide B sligh tly along the inw ard nor mal direction of ∂ F ′ (Ω) at F ′ ( p ) and call it B ′ , so that the p oin t p is no w outside B ′ and y et all the rest of F ′ (Ω) is w ithin B ′ except a small neigh b or ho o d U ′ of p satisfying U ′ ⊂ U . Namely F ′ (Ω) ⊂ B ′ ∪ U ′ with U ′ ⊂ U . Consider the con ve x h ull the union of F ′ (Ω) and B ′ and call it Ω ′ . This domain is con vex and its boundary near F ( p ) coincides the b oundary of F ′ (Ω). Note that the boundary of Ω ′ is ne ither smooth nor strictly conv ex. Ho w eve r it is easy to mo dify Ω ′ sligh tly so that the new ly mo dified domain Ω ′′ is strongly con v ex with C 6 b oundary , and y et the b oundary of Ω ′′ coincides with the b oundary of F ′ (Ω) in a neigh b orho o d of F ′ ( p ). Then one considers Lemp ert discs, i.e., the holomorphic discs that are isometric-and-geo desic embeddings of ∆, for the doma in Ω ′′ . (See [LEM2] for this.) If o ne considers the L emp ert discs cen tered at a p oint sufficien tly close to p with the direction at p nearly complex parallel to a c omplex tangen t direction to the b oundary of Ω ′′ at p , then the image of suc h discs will b e within a small neighborho o d of p . Moreov er suc h Lemp ert discs, sa y h , are also holomorphic geo desic em b eddings of the disc ∆ into Ω, in the sense that ˇ h := [ F ′ ] − 1 ◦ h is the holomorphic geo desic em b edding o f the unit disc ∆ into Ω. F or further details, see the ab ov e cited text in [BUK], esp ecially Prop osition 4.3 and Lemma 4.4 therein. F or con v enience, w e shall call these discs L emp ert-Burns-Kr antz discs for the strongly pseudo conv ex domain Ω, o r LBK-discs fo r short. Suc h discs exist at a p oint sufficien tly close to the b oundary along the directions tha t are approximately complex tang ential to the b oundary . 4.2. H olomorphicit y along the Lemp ert-Burns-Kran t z discs. T ak e now an LBK-disc h : ∆ → Ω in the domain Ω suc h that h ∗ d Ω = d ∆ . Then we first presen t: A KOBA Y AS HI METRIC V ERSION OF BUN WONG’S THEOREM 13 Prop osition 4.1. F or any c ontinuous K ob ayashi distanc e isometry f : Ω → B n , the c omp osition f ◦ h : ∆ → B n is holomorphic o r c onjugate-holomorph i c . Pr o of. Denote b y e h := f ◦ h . W e giv e the pro of in tw o steps. Step 1: The mappin g e h is C ∞ smo oth. W e shall first sho w that e h is smo oth at the o rigin. T ake three p oin ts a, b and c in the unit disc suc h that the Poincar ´ e g eo desic triangle, sa y T ( a, b, c ), with v ertices a t these three p oin ts con tains the origin in its interior. Fill T ( a, b, c ) with the geo desics from a to p oin ts on the geo desic joining b and c . Then ob viously the origin is on one of t hese geo desic s. Now, let m denote the fo ot of t his geo desic. This pro cedure defines a smo oth diffeomorphism, sa y h , from a Euclidean triangle onto T ( a, b, c ), ha ving tw o parameters: one is the t ime parameter of eac h geo desic from the p oint a to a p oin t on the geo desic jo ining b and c , and the other is the pa r a meter o f the geo desic joining b and c . Let e a = e h ( a ) , e b = e h ( b ) and e c = e h ( c ). Let e T ( e a, e b, e c ) b e the geo desic triangle in B n with respect to the P oincar ´ e metric, with v ertices at the three po in ts e a, e b and e c . Again one ma y fill this triangle b y P oincar´ e geo desics of the ball, namely by the geo desics joining e a to the p oints on the g eo desic joining e b and e c . This will aga in yield a smo oth diffeo- morphism fr o m a Euclidean triangle onto the filled tr ia ngle e T ( e a, e b, e c ). Since the Kobay ashi distance-balls are stro ng ly con vex for b oth d ∆ and d B n , it fo llo ws that e T ( e a, e b, e c ) = e h ( T ( a, b, c )). Moreo v er ˜ h maps eac h g e- o desic to a corresponding geo desic with matc hing speed. This sho ws that e f is indeed smo oth at the o rigin. As the argumen t can b e easily mo dified to pro v e the smo othness of e h at an y p o int of the disc ∆, the map e h : ∆ → B is C ∞ smo oth at ev ery p oint. Ac kno wledgemen t: Notice t ha t this arg ument can b e used to giv e a pro o f of the w ell-kno wn theorem of My ers-Steenro d ([MYS]) . This simple but elegan t and p o we rful tec hnique was conv ey ed to the authors b y Rob ert E. Greene in a priv ate comm unication. W e ackno wledge with a great pleasure our indebtedness to him. Step 2. T h e map p ing e h is holomorphic or c onjugate holom o rphic. Since e h maps geo desics to geo desics, it is a geo desic em b edding. Th us the surface e T ( e a, e b, e c ) has the ma ximal holomorphic sectional curv ature, and this can b e realized only by holomorphic sections in the ball. (No- tice that the Koba y ashi metric coincides with the Poincar ´ e metric in the unit ball, and hence it is K¨ ahler with negativ e constant holomorphic 14 KANG-T AE KIM AND STEVEN G. KRANTZ sectional curv ature.) Th us the tangen t plane to the surface e T ( e a, e b, e c ) is complex. Since d ˜ h ∗ ( T ∗ ∆) is alw a ys a complex subspace in T e h ( ∗ ) B n , it follow s by a standard argumen t that e h is either holomorphic or con- jugate holomorphic.  4.3. The Lemp ert map for strongly pseudo c on vex domains. Recall that our domain Ω ′′ is a b ounded strongly conv ex domain with C 6 b oundary . Let x ∈ Ω ′′ . Then, f or a n arbitrarily c hosen z ∈ Ω ′′ with z 6 = x , there exists a unique L emp ert disc h x,z : ∆ → suc h t hat h x,z (0) = x and h x,z ( λ ) = z for some λ with 0 < λ < 1. Then in [LEM1, LEM2] Lemp ert defines Φ( z ) = λh x,z ′ (0) and sho ws that Φ : Ω ′′ → C n extends t o a C 2 smo oth map of the closure Ω ′′ . W e shall call x the pivot of the map Φ. (Although Lemp ert w as mainly in terested in the represen tation map- ping e Φ( z ) = λh x,z ′ (0) / | h x,z ′ (0) | , whic h is now ada ys known as the Lem- p ert represe n tation map e Φ : Ω ′′ → B n , we remark here tha t both Φ and e Φ are kno wn to b e C 2 smo oth up to the b oundary ([LEM2]).) In the next subsection, we will see ho w to use this map Φ to show that the Kobay ashi is ometry f : Ω → B n in question is C 2 up to the b oundary . 4.4. Smo ot h extension of Koba yash i isometry to the b oundary . Cho ose p ′ ∈ S that is sufficien tly close to p . Then let h : ∆ → Ω ′′ b e an LBK-disc with h (0) = F ′ ( p ′ ), as men tioned ab ov e. Let us contin ue to use the notation ˇ h := [ F ′ ] − 1 ◦ h . Then consider t he M¨ obius t r a ns- formation µ : B n → B n whic h maps f ( p ′ ) to the origin. The preceding argumen t s imply that the comp o sition b h := µ ◦ f ◦ ˇ h : ∆ → B n defines a Lemp ert disc at the origin for the unit ball B n . Th us it is linear. More- o v er, it follo ws immediately that b h ( λ ) = λ b h ′ (0) = λ ( µ ◦ f ◦ ˇ h ) ′ (0) = d [ µ ◦ f ] p ′ ( λ ˇ h ′ (0)). It is kno wn that our f , a Kobay ashi distance isometry , admits a Lipsc hitz 1 / 2 extension ([HEN]). But with the strong a ssumption that Ω is b o unded strongly pseudo con v ex with C 6 b oundary , w e shall pr ov e the follow ing: Prop osition 4.2. The Kob ayashi isometry f : Ω → B has a C 2 exten- sion to the b oundary. Mor e pr e cisel y, ther e exists an o p en neighb orh o o d W of ∂ Ω i n C n such that f : Ω ∩ W → B is C 2 smo oth. Pr o of. Notice first that the Kobay ashi isometry f as w ell as f − 1 are lo cally Lipsc hitz with resp ect to the Euclidean metric, as the K oba y ashi A KOBA Y AS HI METRIC V ERSION OF BUN WONG’S THEOREM 15 distance generates the same top ology as the Euclidean distance ([BAR]). Therefore the set S := { x ∈ Ω | d f x and d [ f − 1 ] f ( x ) exist } is a subset of full measure, i.e., the Leb esgue measure of S is the same as t he Leb esgue measure of Ω. In particular, S is dense in Ω. No w let Υ( h ( λ )) := [ d [ µ ◦ f ] p ′ ] − 1 ◦ µ ◦ f ( ˇ h ( λ )) for λ ∈ ∆. Since µ ◦ f ◦ ˇ h ( λ ) = d [ µ ◦ f ] p ′ ( λ ˇ h ′ (0)), it follows that Υ coincides, at ev ery p oin t on the ima g e h (∆), with the aforemen t io ned map Φ : Ω ′′ → C n with its pivot at p ′ . T o b e precise for any ζ ∈ h (∆), let h ( λ ) = ζ . Then Υ( ζ ) = λh ′ (0). Altogether, one sees tha t the mapping [ d [ µ ◦ f ] p ] − 1 ◦ µ ◦ f ◦ [ F ′ ] − 1 coincides with Υ in a small conical neighborho o d (with ap ex at p ) of h ( ∂ ∆) filled b y the LBK -discs at p ′ ; it follows that f is C 2 smo oth in an op en neigh b orho o d of ˇ h ( ∂ ∆). Now it is easy to observ e that this giv es rise to the C 2 smo othness of f in an op en neighborho o d of ∂ Ω as desired.  4.5. The Koba y ashi isometry is CR on the b oundary. W e no w presen t Prop osition 4.3. The r estriction of the extension of f to ∂ Ω into C n is a CR function (or an anti-CR function). Pr o of. Let p ∈ ∂ Ω and let L ∈ T 0 , 1 p ∂ Ω. Regard this v ector field as a deriv ation o p erator on the Euclidean space C n . Then, for ev ery ǫ > 0, there exists an r ∈ (0 , ǫ ) and q ∈ Ω with | p − q | = r suc h that w e may find a Lemp ert disc ϕ : ∆ → Ω satisfying L q f = ∂ ∂ ¯ ζ    0 f ◦ ϕ. Since f ◦ ϕ is holomorphic (replace it b y ¯ f ◦ ϕ if necess ary), o ne immediately sees that L q f = 0 Since f is C 2 up to the b oundar y , letting r tend to zero, one obtains the assertion.  4.6. A nalyticit y of the Koba y ashi isometry – pro of of Theorem 1.3. Finally w e a re ready to presen t: Pr o of of Th e or em 1.3. Start with Prop o sition 4.3 whic h w e just pro v ed. Recall that f restricted to ∂ Ω is a C 2 smo oth CR map from ∂ Ω to ∂ B . 16 KANG-T AE KIM AND STEVEN G. KRANTZ It is also a lo cal diffeomorphism. Notice that f ( ∂ Ω) is compact and relativ ely op en. Therefore f ( ∂ Ω) = ∂ B . This implies that f is a co v ering map. Ho w eve r ∂ Ω is simply connected if n > 1, being top ologically a sphere. Hence f : ∂ Ω → B is a C 2 diffeomorphism. No w apply the Bo chner-Hartogs theorem. The mapping f extends to a holomorphic mapping, sa y b f of Ω into B . Now restrict f to a Lemp ert-Burns-Krantz disc. This has the same v alue as b f at any p o in t of the disc. Therefore, f and b f m ust coincide at ev ery po int of the LBK-disc. Alt o gether, the map f itself is holomorphic in W ∩ Ω for some op en neigh b orho o d W of ∂ Ω. Then one may a sk whether f = b f on Ω. They do coincide indeed. This can b e seen as follows . Since f : Ω → B is a Koba y a shi distance isometry , Ω itself has the prop ert y that any tw o p oints in it mu st ha ve one and the only o ne shortest distance realizing curv e joining t hem. T ak e any suc h curv e γ : [0 , ℓ ] → Ω with γ (0) , γ ( ℓ ) ∈ W ∩ Ω. Then one observ es the follo wing: • d Ω ( γ ( s ) , γ ( t )) = s − t, whenev er 0 ≤ t ≤ s ≤ ℓ . • f ◦ γ is the unique distance realizing curv e j o ining f ( γ (0)) and f ( γ ( ℓ )). • f ( γ (0)) = b f ( γ ( 0)) and f ( γ ( ℓ )) = b f ( γ ( ℓ )). No w for ev ery t ∈ [0 , ℓ ] notice that d Ω ( f ◦ γ (0) , b f ◦ γ ( t )) = d Ω ( b f ◦ γ (0) , b f ◦ γ ( t )) ≤ d Ω ( γ (0) , γ ( t )) = t, and lik ewise d Ω ( b f ◦ γ ( t ) , f ◦ γ ( ℓ )) ≤ ℓ − t. No w b y triangle inequalit y and the uniqueness of the (shortest) dis- tance realizing curv e joining t w o p oin ts, o ne sees immidiately that the inequalities ab ov e are equalities and that b f ( γ ( t )) = f ( γ ( t )) for ev ery t ∈ [0 , ℓ ]. It is no w immediately de duced that f = b f on Ω . In pa r t icular f (and hence b f ) is a bijectiv e holomorphic mapping of Ω on to the ball B . This finishes the pro o f.  Note: Th e authors w ould lik e t o thank H. Sesh adri fo r asking wh ether f can b e sho wn directly to coincide with b f ; w e clarified it changing the end of the pro of slightly . A KOBA Y AS HI METRIC V ERSION OF BUN WONG’S THEOREM 17 5. Concluding Remarks The or ig inal theorem of Bun W ong [W ON], and v arian ts b y R o sa y [R OS] and others, has prov ed to em b o dy a p o w erful and far- reac hing set of ideas. In particular, it w as consideration of this insigh t that led Greene and Kran tz ([GRK2], [GRK 3], [GRK 4]) to form ulate the prin- ciple that the Levi geometry o f a b o undary orbit accum ulat io n p o in t will determine the globa l geometry of the domain. This in turn has led to the Greene-Kran tz conjecture: that a b oundary or bit accumu lation p oin t for a smo o thly b ounded domain m ust in fact b e of finite type in the sense of Kohn-Catlin-D’Ang elo. The result of these studies has b een a profound and fruitf ul dev el- opmen t in geometric analysis. 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Kang-T ae Kim Departmen t of Mathematics P oha ng Unive rsit y of Science and T ech nology P oha ng 79 0-784 The Republic of Ko rea (South) e-mail: kimkt@pos tech.ac.kr Stev en G. Kran tz American Institute of Mathematics 360 Portage Av enue P alo Alto , CA 94306- 2244 U.S.A. e-mail: skrantz@a imath.org