A note on exhaustion of hyperbolic complex manifolds
aa r X i v : . [ m a t h . C V ] J un A NOTE ON EXHAUSTION OF HYPERBOLIC COMPLEXMANIFOLDS
NINH VAN THU , AND TRINH HUY VU Abstract.
The purpose of this article is to investigate a hyperbolic complex manifold M exhausted by a pseudoconvex domain Ω in C n via an exhausting sequence t f j : Ω Ñ M u such that f ´ j p a q converges to a boundary point ξ P B
Ω for some point a P M . introduction Let M and Ω be two complex manifolds. One says that Ω can exhaust M or M canbe exhausted by Ω if for any compact subset K of M there is a holomorphic embedding f K : Ω Ñ M such that f K p Ω q Ą K . In particular, one says that M is a monotone unionof Ω via a sequence of holomorphic embeddings f j : Ω Ñ M if f j p Ω q Ă f j ` p Ω q for all j and M “ Ť j “ f j p Ω q (see [FS77, Fr83]).In [Fr86, Theorem 1], there exists a bounded domain D in C n such that D can exhaustany domain in C n . In addition, the unit ball B n in C n can exhaust many complexmanifods, which are not biholomorphically equivalent to each other (see [For04, FS77]).However, if M in addition is hyperbolic then M must be biholomorphically equivalentto B n (cf. [FS77]). Furthermore, any n -dimensional hyperbolic complex manifold,exhausted by a homogeneous bounded domain D in C n , is biholomorphically equivalentto D . As a consequence, although the polydisc U n and the unit ball B n are bothhomogeneous and there is a domain U in B n that contains almost all of B n , i.e., B n z U hasmeasure zero (cf. [FS77, Theorem 1]) and is biholomorphically equivalent to U n , but U n cannot exhaust the unit ball B n since it is well-known that U n is not biholomorphicallyequivalent to B n .Let M be a hyperbolic complex manifold exhausted by a bounded domain Ω Ă C n via an exhausting sequence t f j : Ω Ñ M u . Let us fix a point a P M . Then, thanks tothe boundedness of Ω, without loss of generality we may assume that f ´ j p a q Ñ p P Ωas j Ñ 8 . If p P Ω, then one always has M is biholomorphically equivalent to Ω (cf.Lemma 2.3 in Section 2).The purpose of this paper is to investigate such a complex manifold M with p P B
Ω.More precisely, our first main result is the following theorem.
Theorem 1.1.
Let M be an p n ` q -dimensional hyperbolic complex manifold and let Ω be a pseudoconvex domain in C n ` with C -smooth boundary. Suppose that M can beexhausted by Ω via an exhausting sequence t f j : Ω Ñ M u . If there exists a point a P M such that the sequence f ´ j p a q converges Λ -nontangentially to a h -extendible boundarypoint ξ P B Ω (see Definition 2.2 in Section 2 for definitions of the Λ -nontangentially Date : June 9, 2020.2010
Mathematics Subject Classification.
Primary 32H02; Secondary 32M05, 32F18.
Key words and phrases.
Hyperbolic complex manifold, exhausting sequence, h -extendible domain. , AND TRINH HUY VU convergence and of the h -extendibility), then M is biholomorphically equivalent to theassociated model M P for Ω at ξ . When ξ is a strongly pseudoconvex boundary point, we do not need the conditionthat the sequence f ´ j p a q converges Λ-nontangentially to ξ as j Ñ 8 . Moreover, inthis circumstance, the model M P is in fact biholomorphically equivalent to M | z | , whichis biholomorphically equivalent to the unit ball B n . More precisely, our second mainresult is the following theorem. Theorem 1.2.
Let M be an p n ` q -dimensional hyperbolic complex manifold and let Ω be a pseudoconvex domain in C n ` . Suppose that B Ω is C -smooth boundary near astrongly pseudoconvex boundary point ξ P B Ω . Suppose also that M can be exhaustedby Ω via an exhausting sequence t f j : Ω Ñ M u . If there exists a point a P M such thatthe sequence η j : “ f ´ j p a q converges to ξ , then M is biholomorphically equivalent to theunit ball B n ` . Notice that Theorem 1.2 is a local version of [DZ19, Theorem 1 .
1] and [Fr83, The-orem I] (see Corollary 3.4 in Section 3). We note that their proofs are based on theboundary estimate of the Fridman invariant and of the squeezing function for stronglypseudoconvex domains. However, in order to prove Theorem 1.1 and Theorem 1.2, weshall use the scaling technique, achieved recently in [Ber06, DN09, NN19].By applying Theorem 1.2 and Lemma 2.3, we also prove that if a hyperbolic complexmanifold M exhausted by a general ellipsoid D P (see Section 4 for the definition of D P ),then M is either biholomorphically equivalent to D P or the unit ball B n (cf. Proposition4.1 in Section 4). In particular, when D P is an ellipsoid E m p m P Z ě q , given by E m “ p z, w q P C : | w | ` | z | m ă ( , in fact Proposition 4.1 is a generalization of [Liu18, Theorem 1].The organization of this paper is as follows: In Section 2 we provide some resultsconcerning the normality of a sequence of biholomorphisms and the h -extendibility. InSection 3, we give our proofs of Theorem 1.1 and Theorem 1.2. Finally, the proof ofProposition 4.1 will be introduced in Section 4.2. The normality and the h -extendibility The normality of a sequence of biholomorphisms.
First of all, we recall thefollowing definition (see [GK87] or [DN09]).
Definition 2.1.
Let t Ω i u i “ be a sequence of open sets in a complex manifold M andΩ be an open set of M . The sequence t Ω i u i “ is said to converge to Ω (writtenlim Ω i “ Ω ) if and only if(i) For any compact set K Ă Ω , there is an i “ i p K q such that i ě i impliesthat K Ă Ω i ; and(ii) If K is a compact set which is contained in Ω i for all sufficiently large i, then K Ă Ω .Next, we recall the following proposition, which is a generalization of the theorem ofH. Cartan (see [DN09, GK87, DT04]). Proposition 2.1.
Let t A i u i “ and t Ω i u i “ be sequences of domains in a complex man-ifold M with lim A i “ A and lim Ω i “ Ω for some (uniquely determined) domains A , NOTE ON EXHAUSTION OF HYPERBOLIC COMPLEX MANIFOLDS 3 Ω in M . Suppose that t f i : A i Ñ Ω i u is a sequence of biholomorphic maps. Supposealso that the sequence t f i : A i Ñ M u converges uniformly on compact subsets of A toa holomorphic map F : A Ñ M and the sequence t g i : “ f ´ i : Ω i Ñ M u convergesuniformly on compact subsets of Ω to a holomorphic map G : Ω Ñ M . Then either ofthe following assertions holds. (i) The sequence t f i u is compactly divergent, i.e., for each compact set K Ă A andeach compact set L Ă Ω , there exists an integer i such that f i p K q X L “ H for i ě i ; or (ii) There exists a subsequence t f i j u Ă t f i u such that the sequence t f i j u convergesuniformly on compact subsets of A to a biholomorphic map F : A Ñ Ω .Remark . By [Ber94, Proposition 2 .
1] or [DN09, Proposition 2 .
2] and by the hypothe-ses of Theorem 1.1 and Theorem 1.2, it follows that for each compact subset K Ť M and each neighborhood U of ξ in C n ` , there exists an integer j “ j p K q such that K Ă f j p Ω X U q for all j ě j . Consequently, the sequence of domains t f j p Ω X U qu converges to M .We will finish this subsection by recalling the following lemma (cf. [Fr83, Lemma1 . Lemma 2.3 (see [Fr83]) . Let M be a hyperbolic manifold of complex dimension n .Assume that M can be exhausted by Ω via an exhausting sequence t f j : Ω Ñ M u , where Ω is a bounded domain in C n . Suppose that there is an interior point a P M such that f ´ j p a q Ñ p P Ω . Then, M is biholomorphically equivalent to Ω . The h -extendibility. In this subsection, we recall some definitions and notationsgiven in [Cat84, Yu95].Let Ω be a smooth pseudoconvex domain in C n ` and p P B
Ω. Let ρ be a local definingfunction for Ω near p . Suppose that the multitype M p p q “ p , m , . . . , m n q is finite.(See [Cat84] for the notion of multitype.) Let us denote by Λ “ p { m , . . . , { m n q .Then, there are distinguished coordinates p z, w q “ p z , . . . , z n , w q such that p “ ρ p z, w q can be expanded near 0 as follows: ρ p z, w q “ Re p w q ` P p z q ` R p z, w q , where P is a Λ-homogeneous plurisubharmonic polynomial that contains no plurihar-monic terms, R is smooth and satisfies | R p z, w q| ď C ˜ | w | ` n ÿ j “ | z j | m j ¸ γ , for some constant γ ą C ą
0. Here and in what follows, a polynomial P is calledΛ-homogeneous if P p t { m z , t { m z , . . . , t { m n z n q “ P p z q , @ t ą , @ z P C n . Definition 2.2 (see [NN19]) . The domain M P “ tp z, w q P C n ˆ C : Re p w q ` P p z q ă u is called an associated model of Ω at p . A boundary point p P B
Ω is called h -extendible if its associated model M P is h -extendible , i.e., M P is of finite type (see [Yu94, Corollary2 . t η j “ p α j , β j qu Ă Ω converges NINH VAN THU , AND TRINH HUY VU Λ -nontangentially to p if | Im p β j q| À | dist p η j , B Ω q| and σ p α j q À | dist p η j , B Ω q| , where σ p z q “ n ÿ k “ | z k | m k . Throughout this paper, we use À and Á to denote inequalities up to a positivemultiplicative constant. Moreover, we use « for the combination of À and Á . Inaddition, dist p z, B Ω q denotes the Euclidean distance from z to B Ω. Furthermore, for µ ą O p µ, Λ q the set of all smooth functions f defined near the origin of C n such that D α D β f p q “ n ÿ j “ p α j ` β j q m j ď µ. If n “ “ p q then we use O p µ q to denote the functions vanishing to order atleast µ at the origin (cf. [Cat84, Yu95]).3. Proofs of Theorem 1.1 and Theorem 1.2
This section is devoted to our proofs of Theorem 1.1 and Theorem 1.2. First of all, letus recall the definition of the Kobayashi infinitesimal pseudometric and the Kobayashipseudodistance as follows:
Definition 3.1.
Let M be a complex manifold. The Kobayashi infinitesimal pseudo-metric F M : M ˆ T , M Ñ R is defined by F M p p, X q “ inf t c ą | D f : ∆ Ñ M holomorphic with f p q “ p, f p q “ X { c u , for any p P M and X P T , M , where ∆ is the unit open disk of C . Moreover, theKobayashi pseudodistance d KM : M ˆ M Ñ R is defined by d KM p p, q q “ inf γ ż F M p γ p t q , γ p t qq dt, for any p, q P M where the infimum is taken over all differentiable curves γ : r , s Ñ M joining p and q . A complex manifold M is called hyperbolic if d KM p p, q q is actually adistance, i.e., d KM p p, q q ą p ‰ q .Next, we need the following lemma, whose proof will be given in Appendix for theconvenience of the reader, and the following proposition. Lemma 3.1.
Assume that t D j u is a sequence of domains in C n ` converging to a model M P of finite type. Then, we have lim j Ñ8 F D j p z, X q “ F M P p z, X q , @p z, X q P M P ˆ C n ` . Moreover, the convergence takes place uniformly over compact subsets of M P ˆ C n ` . Proposition 3.2 (see [NN19]) . Assume that t D j u is a sequence of domains in C n ` converging to a model M P of finite type. Assume also that ω is a domain in C k and σ j : ω Ñ D j is a sequence of holomorphic mappings such that t σ j p a qu Ť M P forsome a P ω . Then t σ j u contains a subsequence that converges locally uniformly to aholomorphic map σ : ω Ñ M P . Now we are ready to prove Theorem 1.1 and Theorem 1.2.
NOTE ON EXHAUSTION OF HYPERBOLIC COMPLEX MANIFOLDS 5
Proof of Theorem 1.1.
Let ρ be a local defining function for Ω near ξ and the multitype M p ξ q “ p , m , . . . , m n q is finite. In what follows, denote by Λ “ p { m , . . . , { m n q .Since ξ is a h -extendible point, there exist local holomorphic coordinates p z, w q in which ξ “ U of 0 as follows:Ω X U “ t ρ p z, w q “ Re p w q ` P p z q ` R p z q ` R p Im w q ` p Im w q R p z q ă u , where P is a Λ-homogeneous plurisubharmonic real-valued polynomial containing nopluriharmonic terms, R P O p , Λ q , R P O p { , Λ q , and R P O p q . (See the proof ofTheorem 1 . .
11 in [Yu95].)By assumption, there exists a point a P M such that the sequence η j : “ f ´ j p a q converges Λ-nontangentially to ξ . Without loss of generality, we may assume that thesequence t η j u Ă Ω X U and we write η j “ p α j , β j q “ p α j , . . . , α jn , β j q for all j . Then,the sequence t η j : “ f ´ p a qu has the following properties:(a) | Im p β j q| À | dist p η j , B Ω q| ;(b) | α jk | m k À | dist p η j , B Ω q| for 1 ď k ď n .For the sequence t η j “ p α j , β j qu , we associate with a sequence of points η j “p α j , . . . , α jn , β j ` ǫ j q , where ǫ j ą
0, such that η j is in the hypersurface t ρ “ u forall j . We note that ǫ j « dist p η j , B Ω q . Now let us consider the sequences of dilations ∆ ǫ j and translations L η j , defined respectively by∆ ǫ j p z , . . . , z n , w q “ ˜ z ǫ { m j , . . . , z n ǫ { m n j , wǫ j ¸ and L η j p z, w q “ p z, w q ´ η j “ p z ´ α j , w ´ β j q . Under the change of variables p ˜ z, ˜ w q : “ ∆ ǫ j ˝ L η j p z, w q , i.e., w ´ β j “ ǫ j ˜ wz k ´ α jk “ ǫ { m k j ˜ z k , k “ , . . . , n, one can see that ∆ ǫ j ˝ L η j p α j , β j q “ p , ¨ ¨ ¨ , , ´ q for all j . Moreover, as in [NN19],after taking a subsequence if necessary, we may assume that the sequence of domainsΩ j : “ ∆ ǫ j ˝ L η j p Ω X U q converges to the following model M P,α : “ tp ˜ z, ˜ w q P C n ˆ C : Re p ˜ w q ` P p ˜ z ` α q ´ P p α q ă u , which is obviously biholomorphically equivalent to the model M P . Without loss ofgenerality, in what follows we always assume that t Ω j u converges to M P .Now we first consider the sequence of biholomorphisms F j : “ T j ˝ f ´ j : M Ą f j p Ω X U q Ñ Ω j , where T j : “ ∆ ǫ j ˝ L η j . Since F j p a q “ p , ´ q and notice that f j p Ω X U q converges to M as j Ñ 8 (see Remark 2.2), by Proposition 3.2, without lossof generality, we may assume that the sequence F j converges uniformly on on everycompact subset of M to a holomorphic map F from M to C n ` . Note that F p M q contains a neighborhood of p , ´ q and F p M q Ă M P .Since F j is normal, by the Cauchy theorem it follows that t J p F j qu converges uniformlyon every compact subsets of M to J p F q , where J p F q denotes the Jacobian determinantof F . However, by the Cartan theorem, J p F j qp z q is nowhere zero for any j because F j is a biholomorphism. Then, the Hurwitz theorem implies that J p F q is a zero function NINH VAN THU , AND TRINH HUY VU or nowhere zero. In the case that J F ” F is regular at no point of M . As F p M q contains a neighborhood of p , ´ q , the Sard theorem shows that F is regular outsidea proper subvariety of M , which is a contradiction. This yields J F is nowhere zero andhence F is regular everywhere on M . By [FS77, Lemma 0], it follows that F p M q isopen and F p M q Ă M P .Next, we shall prove that F is one-to-one. Indeed, let z , z P M be arbitrary. Fix acompact subset L Ť M such that z , z P L . Then, by Remark 2.2 there is a j p L q ą L Ă f j p Ω X U q and F j p L q Ă K Ť M P for all j ą j p L q , where K is a compactsubset of M P . By Lemma 3.1 and the decreasing property of Kobayashi distance, onehas d KM p z , z q ď d Kf j p Ω X U q p z , z q “ d K Ω j p F j p z q , F j p z qqq ď C ¨ d KM P p F j p z q , F j p z qqď C ` d KM P p F p z q , F p z qq ` d KM P p F j p z q , F p z qq ` d KM P p F j p z q , F p z qq ˘ , where C ą j Ñ 8 , we obtain d KM p z , z q ď C ¨ d KM P p F p z q , F p z qq . Since M is hyperbolic, it follows that if F p z q “ F p z q , then z “ z . Consequently, F is one-to-one, as desired.Finally, because of the biholomorphism from M to F p M q Ă M P and the tautness of M P (cf. [Yu95]), it follows that the sequence F ´ j “ f j ˝ T ´ j : T j p Ω X U q Ñ f j p Ω X U q Ă M is also normal. Moreover, since T j ˝ f ´ j p a q “ p , ´ q P M P , it follows that thesequence T j ˝ f ´ j is not compactly divergent. Therefore, by Proposition 2.1, after takingsome subsequence we may assume that T j ˝ f ´ j converges uniformly on every compactsubset of M to a biholomorphism from M onto M P . Hence, the proof is complete. (cid:3) Remark . If M is a bounded domain in C n ` , the normality of the sequence F ´ j canbe shown by using the Montel theorem. Thus, the proof of Theorem 1.1 simply followsfrom Proposition 2.1. Proof of Theorem 1.2.
Let ρ be a local defining function for Ω near ξ . We may assumethat ξ “
0. After a linear change of coordinates, one can find local holomorphiccoordinates p ˜ z, ˜ w q “ p ˜ z , ¨ ¨ ¨ , ˜ z n , ˜ w q , defined on a neighborhood U of ξ , such that ρ p ˜ z, ˜ w q “ Re p ˜ w q ` n ÿ j “ | ˜ z j | ` O p| ˜ w |} ˜ z } ` } ˜ z } q By [DN09, Proposition 3.1] (or Subsection 3 . n “ η in a small neighborhood of the origin, there exists an automorphism Φ η of C n such that ρ p Φ ´ η p z, w qq ´ ρ p η q “ Re p w q ` n ÿ j “ | z j | ` O p| w |} z } ` } z } q . Let us define an anisotropic dilation ∆ ǫ by∆ ǫ p z , ¨ ¨ ¨ , z n , w q “ ˆ z ? ǫ , ¨ ¨ ¨ , z n ? ǫ , wǫ ˙ . NOTE ON EXHAUSTION OF HYPERBOLIC COMPLEX MANIFOLDS 7
For each η P B
Ω, if we set ρ ǫη p z, w q “ ǫ ´ ρ ˝ Φ ´ η ˝ p ∆ ǫ q ´ p z, w q , then ρ ǫη p z, w q “ Re p w q ` n ÿ j “ | z j | ` O p? ǫ q . By assumption, the sequence η j : “ f ´ j p a q converges to ξ . Then, we associate witha sequence of points η j “ p η j , ¨ ¨ ¨ , η jn , η j p n ` q ` ǫ j q , ǫ j ą
0, such that η j is in thehypersurface t ρ “ u . Then ∆ ǫ j ˝ Φ η j p η j q “ p , ¨ ¨ ¨ , , ´ q and one can see that∆ ǫ j ˝ Φ η j pt ρ “ uq is defined by an equation of the formRe p w q ` n ÿ j “ | z j | ` O p? ǫ j q “ . Therefore, it follows that, after taking a subsequence if necessary, Ω j : “ ∆ ǫ j ˝ Φ η p p U ´ q converges to the following domain(1) E : “ t ˆ ρ : “ Re p w q ` n ÿ j “ | z j | ă u , which is biholomorphically equivalent to the unit ball B n ` .Now let us consider the sequence of biholomorphisms F j : “ T j ˝ f ´ j : M Ą f j p Ω X U q Ñ T j p Ω X U q , where T j : “ ∆ ǫ j ˝ Φ η j . Since F j p a q “ p , ´ q , by [DN09, Theo-rem 3.11], without loss of generality, we may assume that the sequence F j convergesuniformly on every compact subset of M to a holomorphic map F from M to C n ` .Note that F p M q contains a neighborhood of p , ´ q and F p M q Ă M P . Following theargument as in the proof of Theorem 1.1, we conclude that F is a biholomorphism from M onto E , and thus M is biholomorphically equivalent to B n ` , as desired. (cid:3) By Lemma 2.3 and Theorem 1.2, we obtain the following corollary, proved by F. S.Deng and X. J. Zhang [DZ19, Theorem 2.4] and by B. L. Fridman [Fr83, Theorem I].
Corollary 3.4.
Let D be a bounded strictly pseudoconvex domain in C n with C -smoothboundary. If a bounded domain Ω can be exhausted by D , then Ω is biholomorphicallyequivalent to D or the unit ball B n . Exhausting a complex manifold by a general ellipsoid
In this section, we are going to prove that if a complex manifold M can be exhaustedby a general ellipsoid D P (see the definition of D P below), then M is biholomorphicallyequivalent to either D P or the unit ball B n .First of all, let us fix n positive integers m , . . . , m n ´ and denote by Λ : “ ´ m , . . . , m n ´ ¯ .We assign weights m , . . . , m n ´ , z , . . . , z n . For an p n ´ q -tuple K “ p k , . . . , k n ´ q P Z n ´ ě , denote the weight of K by wt p K q : “ k ´ ÿ j “ k j m j . Next, we consider the general ellipsoid D P in C n p n ě q , defined by D P : “ tp z , z n q P C n : | z n | ` P p z q ă u , NINH VAN THU , AND TRINH HUY VU where(2) P p z q “ ÿ wt p K q“ wt p L q“ { a KL z K ¯ z L , where a KL P C with a KL “ ¯ a LK , satisfying that P p z q ą z P C n ´ zt u . Wewould like to emphasize here that the polynomial P given in (2) is Λ-homogeneous andthe assumption that P p z q ą z ‰ D P is bounded in C n (cf.[NNTK19, Lemma 6]). Moreover, since P p z q ą z ‰ c , c ą c σ Λ p z q ď P p z q ď c σ Λ p z q , @ z P C n ´ , where σ Λ p z q “ | z | m ` ¨ ¨ ¨ ` | z n ´ | m n ´ . In addition, D P is called a WB-domain if itis strongly pseudoconvex at every boundary point outside the set tp , e iθ q : θ P R u (cf.[AGK16]).Now we prove the following proposition. Proposition 4.1.
Let M be a n -dimensional complex hyperbolic manifold. Suppose that M can be exhausted by the general ellipsoid D P via an exhausting sequence t f j : D P Ñ M u . If D P is a W B -domain, then M is biholomorphically equivalent to either D P orthe unit ball B n .Remark . The possibility that M is biholomorphic onto the unit ball B n is notexcluded because D P can exhaust the unit ball B n by [FM95, Corollary 1 . Proof of Proposition 4.1.
Let q be an arbitrary point in M . Then, thanks to the bound-edness of D P , after passing to a subsequence if necessary we may assume that thesequence t f ´ j p q qu j “ converges to a point p P D P as j Ñ 8 .We now divide the argument into two cases as follows:
Case 1. f ´ j p q q Ñ p P D P . Then, it follows from Lemma 2.3 that M is biholomorphi-cally equivalent to D P . Case 2. f ´ j p q q Ñ p P B D P . Let us write f ´ j p q q “ p a j , a jn q P D P and p “ p a , a n q PB D P . As in [NNTK19], for each j P N ˚ we consider ψ j P Aut p D P q , defined by ψ j p z q “ ˆ p ´ | a jn | q { m p ´ ¯ a jn z n q { m z , . . . , p ´ | a jn | q { m n ´ p ´ ¯ a jn z n q { m n ´ z n ´ , z n ´ a jn ´ ¯ a jn z n ˙ . Then ψ j ˝ f j p q q “ p b j , q , where b j “ ˆ a j p ´ | a jn | q { m , . . . , a j p n ´ q p ´ | a jn | q { m n ´ ˙ , @ j P N ˚ . Without loss of generality, one may assume that b j Ñ b P C n ´ as j Ñ 8 .Since D P is a W B -domain, two possibilities may occur:
Subcase 1: p “ p a , a n q is a strongly pseudoconvex boundary point. In this subcase,it follows directly from Theorem 1.2 that M is biholomorphically equivalent to B n . Subcase 2: p “ p , e iθ q is a weakly pseudoconvex boundary point. In this subcase,one must have a j Ñ and a jn Ñ e iθ as j Ñ 8 . Denote by ρ p z q : “ | z n | ´ ` P p z q adefinition function for D P . Then dist p a j , B D P q « ´ ρ p a j q “ ´ | a jn | ´ P p a j q . Suppose NOTE ON EXHAUSTION OF HYPERBOLIC COMPLEX MANIFOLDS 9 that t a j u converges Λ-nontangentially to p , i.e., P p a j q « σ Λ p a j q À dist p a j , B D P q , orequivalently P p a j q ď C p ´ | a jn | ´ P p a j qq , @ j P N ˚ , for some C ą
0. This implies that P p a j q ď C ` C p ´ | a jn | q , @ j P N ˚ , and thus P p b j q “ ´ | a jn | P p a j q ď C ` C ă , @ j P N ˚ . This yields ψ j ˝ f ´ j p q q “p b j , q Ñ p b, q P D P as j Ñ 8 . So, again by Lemma 2.3 one concludes that M isbiholomorphically equivalent to D P .Now let us consider the case that the sequence t a j u does not converge Λ-nontangentiallyto p , i.e., P p a j q ě c j dist p a j , B D P q , @ j P N ˚ , where 0 ă c j Ñ `8 . This implies that P p a j q ě c j p ´ | a jn | ´ P p a j qq , @ j P N ˚ , for some 0 ă c j Ñ `8 , and hence P p a j q ě c j ` c j p ´ | a jn | q , @ j P N ˚ . Thus, one obtains that P p b j q “ ´ | a jn | P p a j q ě c j ` c j , which implies that P p b q “
1. Consequently, ψ j ˝ f ´ j p q q converges to the strongly pseudoconvex boundary point p “ p b, q of B D P . Hence, as in Subcase 1, it follows from Theorem 1.2 that M isbiholomorphically equivalent to B n .Therefore, altogether, the proof of Proposition 4.1 finally follows. (cid:3) Appendix
Proof of Lemma 3.1.
We shall follow the proof of [Yu95, Theorem 2 .
1] with minor modi-fications. To do this, let us fix compact subsets K Ť M P and L Ť C n ` . Then it sufficesto prove that F D j p z, X q converges to F M P p z, X q uniformly on K ˆ L . Indeed, supposeotherwise. Then, there exist ǫ ą
0, a sequence of points t z j ℓ u Ă K , and a sequence X j ℓ Ă L such that | F D jℓ p z j ℓ , X j ℓ q ´ F M P p z j ℓ , X j ℓ q| ą ǫ , @ ℓ ě . By the homogeneity of the Kobayashi metrics F p z, X q in X , we may assume that } X j ℓ } “ ℓ ě
1. Moreover, passing to subsequences, we may also assumethat z j ℓ Ñ z P K and X j ℓ Ñ X P L as ℓ Ñ 8 . Since M P is taut (see [Yu95,Theorem 3 . p z, X q P M P ˆ C n ` with X ‰
0, there exists an analyticdisc ϕ P Hol p ∆ , M P q such that ϕ p q “ z and ϕ p q “ X { F M P p z, X q . This implies that F M P p z, X q is continuous on M P ˆ C n ` . Hence, we obtain F M P p z j ℓ , X j ℓ q Ñ F M P p z , X q , and thus we have | F D jℓ p z j ℓ , X j ℓ q ´ F M P p z , X q| ą ǫ { ℓ big enough.By definition, for any δ P p , q there exists a sequence of analytic discs ϕ j ℓ P Hol p ∆ , D j ℓ q such that ϕ j ℓ p q “ z , ϕ j ℓ p q “ λ j ℓ X j ℓ , where λ j ℓ ą
0, and F D jℓ p z j ℓ , X j ℓ q ě λ j ℓ ´ δ. , AND TRINH HUY VU It follows from Proposition 3.2 that every subsequence of the sequence t ϕ j ℓ u hasa subsequence converging to some analytic disc ψ P Hol p ∆ , M P q such that ψ p q “ z , ψ p q “ λX , for some λ ą
0. Thus, one obtains that F M P p z , X q ď | ψ p q| for any such ψ . Therefore, one haslim inf ℓ Ñ8 F D jℓ p z j ℓ , X j ℓ q ě F M P p z , X q ´ δ. (4)On the other hand, as in [Yu95], by the tautness of M P , there exists a analytic disc ϕ P Hol p ∆ , M P q such that ϕ p q “ z , ϕ p q “ λX , where λ “ { F M P p z , X q .Now for δ P p , q , let us define an analytic disc ψ δj ℓ : ∆ Ñ C n ` by settings: ψ δj ℓ p ζ q : “ ϕ pp ´ δ q ζ q ` λ p ´ δ qp X j ℓ ´ X q ` p z j ℓ ´ z q for all ζ P ∆ . Since ϕ pp ´ δ q ∆ q is a compact subset of M P and X j ℓ Ñ X , z j ℓ Ñ z as ℓ Ñ 8 , it followsthat ψ δj ℓ p ∆ q Ă D j ℓ for all sufficiently large ℓ , that is, ψ δj ℓ P Hol p ∆ , D j ℓ q . Moreover, byconstruction, ψ δj ℓ p q “ z j ℓ and ` ψ δj ℓ ˘ p q “ p ´ δ q λX j ℓ . Therefore, again by definition,one has F D jℓ p z j ℓ , X j ℓ q ď p ´ δ q λ “ p ´ δ q F M P p z , X q for all large ℓ . Thus, letting δ Ñ ` , one concludes thatlim sup ℓ Ñ8 F D jℓ p z j ℓ , X j ℓ q ď F M P p z , X q . (5)By (4), (5), and (3), we seek a contradiction. Hence, the proof is complete. (cid:3) Acknowledgement.
Part of this work was done while the first author was visiting theVietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thankthe VIASM for financial support and hospitality. The first author was supported by theVietnam National Foundation for Science and Technology Development (NAFOSTED)under grant number 101.02-2017.311.
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Ninh Van Thu Department of Mathematics, Vietnam National University, Hanoi, 334 NguyenTrai, Thanh Xuan, Hanoi, Vietnam Thang Long Institute of Mathematics and Applied Sciences, Nghiem Xuan Yem,Hoang Mai, HaNoi, Vietnam
E-mail address : [email protected] Trinh Huy Vu Department of Mathematics, Vietnam National University at Hanoi, 334 NguyenTrai str., Hanoi, Vietnam
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