Boundary behavior of the squeezing function near a global extreme point
BBOUNDARY BEHAVIOR OF THE SQUEEZING FUNCTION NEARA GLOBAL EXTREME POINT
NINH VAN THU , , NGUYEN THI KIM SON , AND CHU VAN TIEP Abstract.
In this paper, we prove that the general ellipsoid D P is holomorphicallyhomogeneous regular provided that it is a W B -domain. Then, the uniform lowerbound for the squeezing function near a p P, r q -extreme point is also given. introduction Let Ω be a domain in C n and p P Ω. For a holomorphic embedding f : Ω Ñ B n : “ B p
0; 1 q with f p p q “
0, we set σ Ω ,f p p q : “ sup t r ą B p r q Ă f p Ω qu , where B p z ; r q Ă C n denotes the ball of radius r with center at z . Then the squeezingfunction σ Ω : Ω Ñ R is defined in [DGZ12] as σ Ω p p q : “ sup f t σ Ω ,f p p qu . Notice that 0 ă σ Ω p z q ď z P Ω and the squeezing function is obviously invariantunder biholomorphisms. A domain Ω in C n is called holomorphically homogeneousregular (HHR) if inf z P Ω σ Ω p z q ą C -smooth bounded strongly pseudoconvex domains (see e.g., [DWZZ20] and thereferences therein).As a main result of this paper, we first present a new class of HHR domains, forinstance, a nonconvex domain tp z , z q P C : | z | ` | z | ` | z | Re p z q{ ă u can be itstypical element. To do this, for positive integers m , . . . , m n ´ let us consider a generalellipsoid 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 , where P p z q is a p { m , . . . , { m n ´ q -homogeneous polynomial given by 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 ‰
0. Here andin what follows, z : “ p z , . . . , z n ´ q and wt p K q : “ ř n ´ j “ k j m j denotes the weight of any Mathematics Subject Classification.
Primary 32H02; Secondary 32M17, 32F45.
Key words and phrases.
Squeezing function, HHR domain, automorphism group, general ellipsoid. a r X i v : . [ m a t h . C V ] J un NINH VAN THU, NGUYEN THI KIM SON, AND CHU VAN TIEP multi-index K “ p k , . . . , k n ´ q P N n ´ with respect to Λ : “ p { m , . . . , { m n ´ q . (SeeSection 3 for the definition of weighted homogeneous polynomials.) The domain D P iscalled a WB-domain if D P is strongly pseudoconvex at every boundary point outsidethe set tp , e iθ q : θ P R u (cf. [AGK16]).Our first main result is the following theorem. Theorem 1.1.
Let P be a weighted homogeneous polynomial with weight p m , . . . , m n ´ q given by (1) such that P p z q ą for all z P C n ´ zt u . If D P is a WB-domain, then D P is HHR. Thanks to the explicit description for the automorphism group of D P , denoted byAut p D P q , given in [NNTK19], we shall deduce the lower estimation of the squeezingfunction of D P to the subset tp z , q P D P : P p z q “ u , whose closure intersects B D P atonly strongly pseudoconvex boundary points. Then our proof of Theorem 1.1 becomesa simple consequence of Theorem 2.3 as shown in Section 2.As a consequence of Theorem 1.1, by [DGZ16, LSY04, Ye09], one obtains the followingcorollary. Corollary 1.2.
The Carath´eodory metric, the Kobayashi metric, the Bergman metricand the K¨ahler-Einstein metric on D P are all equivalent and complete, provided that D P is a WB-domain. Next, as our second main result, we shall discuss the behavior of the squeezing func-tion near a global extreme point. Let Ω be a domain in C n with C -smooth boundarynear a boundary point p P B
Ω. Let us recall that a boundary point p is said to be spherically extreme if there exists a ball B p c p p q ; R q in C n of some radius R , centeredat some point c p p q such that Ω Ă B p c p p q ; R q and p P B Ω X B B p c p p q ; R q (see [KZ16]).It was shown that lim Ω Q q Ñ p σ Ω p q q “ p is a spherically extreme boundary point ofΩ (see [KZ16, Theorem 3 . p , q P B D P is not spherically extreme if min t m , . . . , m n ´ u ě Ă C n , which is locally convexifi-able and has finite 1-type 2 k at p P B
Ω, having a Stein neighborhood basis, there existsa holomorphic embedding f : Ω Ñ C n such that f p p q is a global extreme point of type 2 k for f p Ω q in the sense that f p Ω q Ă B nk : “ t z P C n : | z n | ` } z } k ă u and f p p q “ p , q .Therefore, this motivates us to introduce the following definition. Definition 1.1.
Let Ω be a domain in C n ( n ě p P B Ω, P p z q be the polynomialgiven in (1), and r P p , s . We say that p is p P, r q -extreme if there exists a holomorphicembedding f : Ω Ñ E P such that f p p q “ p , q and D p r q Ă f p Ω q , where E P : “ tp z , z n q P C n : P p z q ă p z n qu ; D p r q : “ D P,r “ tp z , z n q P C n : | z n ´ r | ` P p z q ă r u . In this situation, denote by Γ p r, c q : “ f ´ p D p r q X tp z , z n q P C n : | Im p z n q| ď c | Re p z n q|u )for c ą Remark . Since D p r q Ă D p r q for 0 ă r ă r ď
1, it follows that if p is p P, r q -extreme,then it is also p P, r q -extreme for any 0 ă r ă r ď
1. In addition, p is sphericallyextreme if and only if it is p} z } , r q -extreme for some 0 ă r ď OUNDARY BEHAVIOR OF THE SQUEEZING FUNCTION 3
Our second main result is the following theorem.
Theorem 1.3.
Let Ω be a domain in C n p n ě q and p P B Ω be a p P, r q -extreme pointwith ă r ď . Then, for any ă r ă r and c ą there exist (cid:15) , γ ą such that σ Ω p q q ą γ , @ q P Γ p r , c q X B p p ; (cid:15) q . Remark . The convergence of a sequence of points in Γ p r, c q to p is exactly the Λ-nontangential convergence introduced in [NN19, Definition 3 . t a j u Ă f ´ p D p r qq Ă Ω does not converge Λ-nontangentially to p “
0, i.e., for any0 ă r ă r and c ą j r ,c P N such that a j R Γ p r , c q for all j ě j r ,c , wedo not know the behavior of t σ Ω p a j qu . Therefore, a natural question to ask is whetherlim inf f ´ p D p r qqQ z Ñ p σ Ω p z q ą p D P q given in [NNTK19] and the proof of Theorem 1.1is given. Then, we shall prove Theorem 1.3 in detail in Section 4. In addition, analternative proof of Theorem 1.1 will be presented in Appendix.2. Several properties of the squeezing function
Let Ω be a bounded domain in C n . Denote r p z, Ω q and R p z, Ω q , respectively, by r p z, Ω q “ sup t r ą B p z ; r q Ă Ω u and R p z, Ω q “ inf t R ą B p z ; R q Ą Ω u . For a subset K Ă Ω, we put r p K, Ω q “ inf z P K r p z, Ω q ; R p K, Ω q “ sup z P K R p z, Ω q . Now we prepare a technical lemma.
Lemma 2.1.
Let Ω be a bounded domain in C n and K be a relative compact subset of Ω .Then one has r p z, Ω q{ R p z, Ω q ď σ Ω p z q ď for any z P Ω and inf z P K σ Ω p z q ě r p K, Ω q R p K, Ω q ą .Proof. Let us consider an affine map f p ζ q : “ ζ ´ zR p z, Ω q . Then one sees that f is abiholomorphic map from Ω into B n . Moreover, we have B p r p z, Ω q{ R p z, Ω qq Ă f p B p z ; r p z, Ω qqq Ă f p Ω q , and hence inf z P K σ Ω p z q ě inf z P K r p z, Ω q R p z, Ω q “ r p K, Ω q R p K, Ω q ą . Therefore, the proof is complete. (cid:3)
Definition 2.1.
Let Ω be a bounded domain in C n and Σ be a subset of Ω. We saythat Ω is HHR on
Σ if inf z P Σ σ Ω p z q ą
0. In particular, Ω is
HHR if it is HHR on Ω.Next, we prepare a proposition which is crucial in our proof of Theorem 1.1.
Proposition 2.2.
Let Ω be a bounded domain in C n . Suppose that there exists a subset Σ Ă Ω satisfying that @ z P Ω D f P Aut p Ω q such that f p z q P Σ . Then Ω is HHR if it isHHR on Σ . NINH VAN THU, NGUYEN THI KIM SON, AND CHU VAN TIEP
Proof.
Suppose that Ω is HHR on Σ. By definition, there exists c ą σ Ω p z q ě c for all z P Σ. Now let z P Ω be arbitrary. We shall prove that σ Ω p z q ě c .Indeed, by assumption there exists an automorphism f P Aut p Ω q such that f p z q P Σand thus σ Ω p z q “ σ Ω p f p z qq ě c because of the invariance of the squeezing functionunder biholomorphisms. Hence, the proof is complete. (cid:3) Let us recall that a compact set K Ť C n is said to have a Stein neighborhood basisif for any domain V containing K there exists a pseudoconvex domain Ω V such that K Ă Ω V Ă V . For instance, the closure Ω of a smooth bounded pseudoconvex domainΩ in C n has a (strong) Stein neighborhood basis if Ω has a defining function ρ and thereexists (cid:15) ą t z P C n : ρ p z q ă (cid:15) u is pseudoconvex for all (cid:15) P r , (cid:15) s (cf. [Sa12]).Recently, the behavior of the squeezing function near a strongly pseudoconvex bound-ary point was established. Theorem 2.3 ([DGZ16, DFW14, KZ16]) . If a bounded domain Ω in C n admittinga Stein neighborhood basis has a strongly pseudoconvex boundary point, say p , then lim z Ñ p σ Ω p z q “ . As a consequence, Theorem 2.3, together with Lemma 2.1 and Proposition 2.2, impliesthe following:
Corollary 2.4.
Let Ω be a bounded domain in C n admitting a Stein neighborhood basis.Suppose that there exists a subset M Ă Ω satisfying that @ z P Ω D f P Aut p Ω q such that f p z q P M . If each p P M X B Ω is strongly pseudoconvex, then Ω is HHR.Proof. By Proposition 2.2, it suffices to show that Ω is HHR on M . Indeed, supposeotherwise that there exists a sequence t z j u Ă M such that σ Ω p z j q Ñ j Ñ 8 . Takinga subsequence if necessary we may assume that either z j Ñ p P M X B
Ω or t z j u Ť Ω.By virtue of Lemma 2.1, the latter case does not occur. On the other hand, for theformer case one has σ Ω p z j q Ñ j Ñ 8 by Theorem 2.3, which is a contradiction.This ends the proof. (cid:3)
Remark . By Lemma 2.1, it is easy to see that Ω is HHR on any relative compactsubset K Ť Ω. In addition, one can infer that if σ Ω p p q “ p P Ω, then Ω isbiholomorphic to the unit open ball (cf. [DGZ12]). Moreover, it follows from Corollary2.4 that Ω is HHR if each p P Ω { Aut p Ω q X B Ω is strongly pseudoconvex.3.
Squeezing function of the general ellipsoid
Let us assign weights m , . . . , m n ´ , z , . . . , z n ´ , z n , respectivelyand denote by wt p K q : “ ř n ´ j “ k j m j the weight of an p n ´ q -tuple K “ p k , . . . , k n ´ q P Z n ´ ě . A real-valued polynomial P on C n ´ is called a weighted homogeneous polynomialwith weight p m , . . . , m n ´ q (or simply p { m , . . . , { m n ´ q -homogeneous ), if P p t { m z , . . . , t { m n ´ z n ´ q “ tP p z , . . . , z n ´ q for all z P C n ´ and t ą . In the case when m “ m “ ¨ ¨ ¨ “ m n ´ , then P is called homogeneous of degree m .We note that if P p z q is a p { m , . . . , { m n ´ q -homogeneous polynomial, then P p z q “ ÿ wt p K q` wt p L q“ a KL z K ¯ z L , OUNDARY BEHAVIOR OF THE SQUEEZING FUNCTION 5 where a KL P C with a KL “ ¯ a LK (see [NNTK19]).Throughout this paper, let P p z q be a p { m , . . . , { m n ´ q -homogeneous polynomialgiven by(1) 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 ‰ . In addition,since P p z q ą z ‰ and by the weighted homogeneity, there are two constants c , c ą c σ Λ p z q ď P p z q ď c σ Λ p z q , where σ Λ p z q “ ř n ´ j “ | z j | m j (cf. [NNTK19, Lemma 6], [Yu95]). Furthermore, one seesthat D P has a Stein neighborhood basis.We first consider the general ellipsoid D P and the model E P in C n p n ě q , definedrespectively by D P : “ tp z , z n q P C n : | z n | ` P p z q ă u ; E P : “ tp z , z n q P C n : P p z q ă p z n qu . Then, we need the following lemma which is essentially well-known (cf. [BP94]).
Lemma 3.1.
Let P be a weighted homogeneous polynomial with weight p m , . . . , m n ´ q given by (1) such that P p z q ą for all z P C n ´ zt u . Then, the holomorphic map ψ defined by p z , z n q ÞÑ ˆ { m p ` z n q { m z , . . . , { m n ´ p ` z n q { m n ´ z n ´ , ´ z n ` z n ˙ , is a biholomorphism from D P onto E P .Proof. Indeed, a direct computation shows thatRe ˆ ´ z n ` z n ˙ “ Re ˆ p ´ z n qp ` ¯ z n q| ` z n | ˙ “ ´ | z n | | ` z n | . Moreover, since P has the form as in (1), it follows that P ˆ { m p ` z n q { m z , . . . , { m n ´ p ` z n q { m n ´ z n ´ ˙ “ | ` z n | P p z q . Therefore, one can deduce that2Re ˆ ´ z n ` z n ˙ ´ P ˆ { m p ` z n q { m z , . . . , { m n ´ p ` z n q { m n ´ z n ´ ˙ ą | z n | ` P p z q ă . Hence, the conclusion easily follows from the previous relation. (cid:3)
Remark . A direct computation also shows that ψ ´ “ ψ , ψ p , q “ p , q and ψ p , q “ p , q . In addition, ψ p z q Ñ p , ´ q as E P Q z Ñ 8 .Next, the first author et al. [NNTK19] pointed out the following lemma.
NINH VAN THU, NGUYEN THI KIM SON, AND CHU VAN TIEP
Lemma 3.2 (see Lemma 7 in [NNTK19]) . Let P be a weighted homogeneous polynomialwith weight p m , . . . , m n ´ q given by (1) such that P p z q ą for all z P C n ´ zt u . Then, Aut p D P q contains the following automorphisms φ a,θ , defined by (2) p z , z n q ÞÑ ˆ p ´ | a | q { m p ´ ¯ az n q { m z , . . . , p ´ | a | q { m n ´ p ´ ¯ az n q { m n ´ z n ´ , e iθ z n ´ a ´ ¯ az n ˙ , where a P ∆ : “ t z P C : | z | ă u and θ P R . In addition, Aut p E P q contains the dilations Λ λ , λ ą , defined by Λ λ p z , z n q “ ´ z λ { m , . . . , z n ´ λ { m n ´ , z n λ ¯ . Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1.
Denote by Σ : “ tp z , q P D P : P p z q “ u . Then for each p “p p , p n q P D P , one has φ p n , p p q P Σ, where φ p n , is an automorphism given in (2). Sinceeach boundary point of Σ X B D P is strongly pseudoconvex and the asymptotic limitvalue of the squeezing function is 1 by Theorem 2.3, it follows that σ D P p z q is uniformlybounded away from zero on Σ. Therefore, the assertion finally follows by Proposition2.2. (cid:3) Proof of Theorem 1.3
This section is fully devoted to the proof of Theorem 1.3. Let Ω be a domain describedin the hypothesis of Theorem 1.3. That is, p P B
Ω is a p P, r q -extreme point with 0 ă r ď
1. Then, by virtue of the invariance of the squeezing function under biholomorphisms,we may assume that D p r q Ă Ω Ă E P and p “ p , q . Let us fix 0 ă r ă r , c ą
0. ThenΓ p r , c q becomes Γ p r , c q “ D p r q X tp z , z n q P C n : | Im p z n q| ď c | Re p z n q|u . ψ ˝ Λ λ p Ω q w z Re z n r Σ E r E r ED p r q Ω D p r q r rψ “ ψ ´ D P D r D r Σ 1 w n r Σ “ ψ p Σ q D r “ ψ p E r q Figure 1.
OUNDARY BEHAVIOR OF THE SQUEEZING FUNCTION 7
Let us denote by E r : “ E P { r , E : “ E “ E P , D r : “ tp z , z n q P C n : | z n | ` P p z q{ r ă u , D : “ D “ D P for simplicity. Then one can see that D p r q Ă D p r q Ă Ω Ă E and D p r q Ă E r . Moreover, the biholomorphism ψ , given in Lemma 3.1, maps E, E r , E r onto D, D r , D r , respectively (see Lemma 3.1 and Figure 1 above). Furthermore, byLemma 3.2, E, E r , E r are invariant under the action of Λ λ , defined byΛ λ p z q : “ ´ z λ { m , . . . , z n ´ λ { m n ´ , z n λ ¯ , λ ą . In addition, a straightforward calculation shows thatΛ λ p D p r qq “ (cid:32) p z , z n q P C n : λ | z n | ` P p z q ă r Re p z n q ( , which converges to E r as λ Ñ ` . Moreover, ψ ˝ Λ λ p D p r qq converges to D r as λ Ñ ` .Here and in what follows, a family of domains t X λ u λ P R ` is said to converge to a domain X as λ Ñ ` if for each compact subset K Ť X , there exists λ “ λ p K q such that K Ă X λ for all 0 ă λ ă λ .Now let us define a set r Σ : “ ! z “ p z , z n q P E r : } z } “ , | Im p z n q| ď c | Re p z n q| ) “ t z “ p z , z n q P C n : P p z q ă r Re p z n q , } z } “ , | Im p z n q| ď c | Re p z n q|u Ť E r and then Σ : “ ψ p r Σ q Ă D r . Then, one has r Σ Ť E r , and thus Σ Ť D r . In particular,for any q P E r with | Im p q n q| ă c | Re p q n q| , the orbit t Λ λ p q qu λ P R ` meets the set r Σ in a(unique) point.Let q P Γ p r , c q X B p (cid:15) q Ă E r X B p (cid:15) q be arbitrary, where (cid:15) ą λ ą λ p q q P r Σ, i.e., } Λ λ p q q} “
1. Notice that λ Ñ ` whenever q Ñ p “ p , q . In addition, since ψ ˝ Λ λ p D p r qq converges to D r as λ Ñ ` , it follows that (cid:15) ą Ť ψ ˝ Λ λ p D p r qq Ă ψ ˝ Λ λ p Ω q anddist ` Σ , B ` ψ ˝ Λ λ p Ω q ˘˘ ě dist ` Σ , B ` ψ ˝ Λ λ p D p r q ˘˘ ą dist p Σ , B D r q{ ą q P D p r q X B p (cid:15) q . Here and in what follows, dist p A, B q denotes the Euclideandistance between two subsets A, B Ă C n .In summary, the biholomorphism G λ : “ ψ ˝ Λ λ from E onto D satisfies the followingproperties:a) G λ p Ω q Ă D ;b) G λ p q q P Σ Ť D r ;c) dist ` G λ p q q , B ` G λ p Ω q ˘˘ ą δ ,where δ : “ dist p Σ , B D r q{
2. Therefore, by Lemma 2.1 and again by the invariance of thesqueezing function under biholomorphisms, we conclude that σ Ω p q q “ σ G λ p Ω q p G λ p q qq ą δ { d ą , @ q P Γ p r , c q X B p (cid:15) q , where d denotes the diameter of D P . This ends the proof of Theorem 1.3 with γ : “ δ { d . l NINH VAN THU, NGUYEN THI KIM SON, AND CHU VAN TIEP
Appendix
In this Appendix, we shall present an alternative proof of Theorem 1.1 by the followingargument. Indeed, suppose otherwise that there exists a sequence t a j “ p a j , a jn qu Ă D P such that a j Ñ p P B D P and σ D P p a j q Ñ j Ñ 8 . If p is strongly pseudoconvex, then σ D P p a j q Ñ j Ñ 8 by Theorem 2.3. Hence our conclusion immediately follows fromthis case. Moreover, since D P is a WB-domain, it suffices to assume that p “ p , e iθ q , θ P R , which is weakly pseudoconvex. In this case, 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 a local defining function for D P . Thendist p a j , B D P q « ´ ρ p a j q « ´ | a jn | ´ P p a j q . Here and in what follows, we use symbols À and Á to denote inequalities up to a positive multiplicative constant. In addition, weuse a symbol « for the combination of À and Á .Let us denote by ψ j : “ φ a jn , the automorphism of D P given in Lemma 3.2. Then ψ j P Aut p D P q , given 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 ˙ , and hence ψ j p a j 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 ´ ˙ . Thanks to the boundedness of t b j u , without loss of generality we may assume that b j Ñ b P C n ´ as j Ñ 8 .We now divide the argument into two cases as follows:
Case 1.
The sequence t a j u converges Λ-nontangentially to p (cf. Remark 2). Then P p a j q À dist p a j , B D P q . Without loss of generality, we may assume that P p a j q ď C p ´| a jn | ´ P p a j qq for some C ą
0. This implies that P p a j q ď C ` C p ´ | a jn | q . Hence, P p b j q “ ´ | a jn | P p a j q ď C ` C ă ψ j p a j q “ p b j , q Ñ p b, q P D P with P p b q ă
1. Therefore, Lemma 2.1 yields lim inf j Ñ8 σ D P p a j q ą
0, which is absurd.
Case 2.
The sequence t a j u does not converge Λ-nontangentially to p . Then P p a j q ě c j dist p a j , B D P q for some sequence t c j u Ă R with 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 for some sequence t c j u Ă R with 0 ă c j Ñ `8 and hence P p a j q ě c j ` c j p ´ | a jn | q , @ j ě . This tells us that P p b j q “ ´ | a jn | P p a j q ě c j ` c j , @ j ě
1. Therefore, we arrive at thesituation b j Ñ b with P p b q “ ψ j p a j q converges to the strongly pseudoconvexboundary point p b, q of B D P , which implies that σ D P p a j q “ σ D P p ψ j p a j qq Ñ j Ñ 8 again by Theorem 2.3. This leads to a contradiction.Hence, altogether, we complete the proof of Theorem 1.1.
OUNDARY BEHAVIOR OF THE SQUEEZING FUNCTION 9
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 third author was supported bythe Vietnam National Foundation for Science and Technology Development (NAFOS-TED) under grant number 101.99-2019.326. It is a pleasure to thank Hyeseon Kim forstimulating discussion.
References [AGK16] T. Ahn, H. Gaussier, and K.-T. Kim,
Positivity and completeness of invariant metrics , J.Geom. Anal. (2016), no. 2, 1173–1185.[BP94] E. Bedford and S. Pinchuk, Convex domains with noncompact groups of automorphisms , Mat.Sb. (1994), no. 5, 3–26; translation in Russian Acad. Sci. Sb. Math. 82 (1995), no. 1, 1–20.[DGZ12] F. Deng, A. Guan, and L. Zhang,
Some properties of squeezing functions on bounded domains ,Pacific J. Math. (2012), no. 2, 319–341.[DGZ16] F. Deng, A. Guan, and L. Zhang,
Properties of squeezing functions and global transformationsof bounded domains , Trans. Amer. Math. Soc. (2016), no. 4, 2679–2696.[DWZZ20] F. Deng, Z. Wang, L. Zhang and X. Zhou,
Holomorphic Invariants of bounded domains , J.Geom. Anal. (2020), no. 2, 1204–1217.[DFW14] K. Diederich, J. E. Fornæss, and E. F. Wold, Exposing points on the boundary of a strictlypseudoconvex or a locally convexifiable domain of finite -type , J. Geom. Anal. (2014), 2124–2134.[KZ16] K.-T. Kim and L. Zhang, On the uniform squeezing property and the squeezing function , Pac.J. Math. (2016), no. 2, 341–358.[LSY04] K. Liu, X. Sun, and S.-T. Yau,
Canonical metrics on the moduli space of Riemann surfaces ,I. J. Differential Geom. (2004), no. 3, 571–637.[NN19] V. T. Ninh and Q. D. Nguyen, Some properties of h-extendible domains in C n ` , J. Math.Anal. Appl. (2020), no. 2, 123810, 14 pp.[NNTK19] V. T. Ninh, T. L. H Nguyen, Q. H. Tran, and H. Kim, On the automorphism groups offinite multitype models in C n , J. Geom. Anal. (2019), no. 1, 428–450.[Sa12] S. S¸ahutoˇglu, Strong Stein neighbourhood bases , Complex Var. Elliptic Equ. (2012), no. 10,1073–1085.[Ye09] S.-K. Yeung, Geometry of domains with the uniform squeezing property , Adv. Math. (2009),no. 2, 547–569.[Yu95] J. Yu,
Weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseu-doconvex domains , Trans. Amer. Math. Soc. (1995), no. 2, 587–614.
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] Nguyen Thi Kim Son Department of Mathematics, Hanoi University of Mining and Geology, 18 Pho Vien,Bac Tu Liem, Hanoi, Vietnam
E-mail address : [email protected] Chu Van Tiep Department of Mathematics, The University of Danang - University of Scienceand Education, 459 Ton Duc Thang, Danang, Vietnam.
E-mail address ::