This paper introduces the notion of squeezing functions on bounded domains and studies some of their properties. The relation to geometric and analytic structures of bounded domains will be investigated. Existence of related extremal maps and continuity of squeezing functions are proved. Holomorphic homogeneous regular domains introduced by Liu, Sun and Yau are exactly domains whose squeezing functions have positive lower bounds. Completeness of certain intrinsic metrics and pseudoconvexity of holomorphic homogeneous regular domains are proved by alternative method. In the dimension one case, we get a neat description of boundary behavior of squeezing functions of finitely connected planar domains. This leads to necessary and sufficient conditions for a finitely connected planar domain to be a holomorphic homogeneous regular domain. Consequently, we can recover some important results in complex analysis. For annuli, we obtain some interesting properties of their squeezing functions. Finally, we present some examples of bounded domains whose squeezing functions can be given explicitly.
The central purpose of the present paper is to study boundary behavior of squeezing functions on bounded domains. We prove that the squeezing function of a strongly pseudoconvex domain tends to 1 near the boundary. In fact, such an estimate is proved for the squeezing function on any domain near its globally strongly convex boundary points. We also study the stability of squeezing functions on a sequence of bounded domains, and give comparisons of intrinsic measures and metrics on bounded domains in terms of squeezing functions. As applications, we give new and simple proofs of several well known results about geometry of strongly pseudoconvex domains, and prove that all Cartan-Hartogs domains are homogenous regular. Finally, some related problems that ask for further study are proposed. introductionIn a recent work [5], the authors introduced the notion of squeezing functions to study geometric and analytic properties of bounded domains. The squeezing function of a bounded domain D is defined as follows: Definition 1.1. Let D be a bounded domain in C n . For z ∈ D and an (open) holomorphic embedding f : D → B n with f (z) = 0, we define s D (z, f ) = sup{r|B n (0, r) ⊂ f (D)}, and the squeezing number s D (z) of D at z is defined aswhere the supremum is taken over all holomorphic embeddings f : D → B n with f (z) = 0, B n is the unit ball in C n , and B n (0, r) is the ball in C n with center 0 and radius r. As z varies, we get a function s D on D, which is called the squeezing function of D.Roughly speaking, s D (z) describes how does the domain D look like the unit ball, observed at the point z. By definition, it is clear that squeezing functions are invariant under biholomorphic transformations. Namely, if f : D 1 → D 2 is a holomorphic equivalence of two bounded domains, then s D2 • f = s D1 . Though the definition is so simple, it is turned out that so many geometric and analytic properties of bounded domains are encoded in their squeezing functions.
We prove that the bounded convex domains and the C 2 -smoothly bounded strongly pseudoconvex domains in ރ n admit the uniform squeezing property. Moreover, we prove by the scaling method that the squeezing function approaches 1 near the strongly pseudoconvex boundary points.
Abstract. We study the complete Kähler-Einstein metric of a Hartogs domain Ω built on an irreducible bounded symmetric domain Ω, using a power N µ of the generic norm of Ω. The generating function of the Kähler-Einstein metric satisfies a complex Monge-Ampère equation with boundary condition. The domain Ω is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by X ∈ [0, 1[. This allows to reduce the Monge-Ampère equation to an ordinary differential equation with limit condition. This equation can be explicitly solved for a special value µ 0 of µ. We work out the details for the two exceptional symmetric domains. The special value µ 0 seems also to be significant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.
Background: MicroRNAs (miRNAs) have been shown to contribute to the initiation and progression of human cancer, including retinoblastoma. However, expression levels and potential roles of miRNAs in retinoblastoma remain largely unknown. In this study, we aimed to identify dysregulated miRNAs and explore their functional roles in the development of retinoblastoma. Material and Methods: First, miRNA expression profiling in retinoblastoma tissues was performed via microarray analysis. To evaluate the involvement of miR-214-3p in multi-drug resistance, gain-of-function experiments were employed in vitro and in vivo. Bioinformatics analysis, luciferase reporter assay, qRT-PCR and Western blot were used to investigate the underlying mechanisms. Results: Here, we identified 57 up-regulated and 34 down-regulated miRNAs. Among them, miR-214-3p was the most significantly decreased. We found that miR-214-3p level was positively correlated with clinical outcome and chemotherapy response. Overexpression of miR-214-3p significantly sensitized retinoblastoma cells to multiple chemodrugs and promoted cell apoptosis in vitro and in vivo. Further investigations revealed that miR-214-3p directly regulated ABCB1 and XIAP expression through interacting with the 3' untranslated regions (3'UTRs). Pearson correlation analysis showed that miR-214-3p expression in retinoblastoma tissues was negatively correlated with ABCB1 and XIAP expression. We also observed that overexpression of ABCB1 or XIAP partly reversed the chemoresistance inhibition-induced by miR-214-3p overexpression. Conclusion: Our data demonstrate that miR-214-3p functions as a tumor suppressor to inhibit the chemoresistance in retinoblastoma, suggesting that miR-214-3p might be potential diagnostic and therapeutic targets for retinoblastoma treatment.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.