粗糙拟映射和Gromov双曲性

本项目计划主要研究以下内容。（1）度量空间上Gromov双曲性的特征：2003年，Balough等在研究Gromov双曲性、Gehring-Hayman不等式以及球可分离性的联系时提出了关于Gehring-Hayman不等式和球可分离性关系的公开问题，此问题的肯定回答将给出Gromov双曲性新的几何刻画。我们将采用几何与分析相结合的方法来开展研究。（2）度量空间上Gromov双曲性与粗糙拟映射：2001年，Bonk等建立了欧几里得空间中的Gromov双曲性与拟共形映射等之间的关系。我们将在Bonk-Schramm粗糙理论的基础上，通过构造新的映射类来揭示度量空间中Gromov双曲空间与粗糙拟映射等之间的内在联系。（3）度量空间上拟等距映射的特征及应用：我们将建立相对拟双曲度量的拟等距映射的特征，并计划利用此特征来研究Vaisala于1999年提出的相关公开问题。此研究具有重要的理论意义。

The main purpose of this project is to study the following problems...(1) Characterization of Gromov hyperbolicity in metric spaces: In 2003, when Balough and Buckney discussed the relationships among Gromov hyperbolicity, Gehring-Hayman inequality and Ball separation property, they raised an open problem on the relationship between Gehring-Hayman inequality and Ball separation property. An affirmative answer of this open problem will give a new geometric characterization for Gromov hyperbolicity. By combining a geometric method and an analytical approach, we shall work on this problem. ..(2) Gromov hyperbolicity and coarse quasi-mappings in metric spaces: In 2001, Bonk, Heinonen and Koskela established a relation of Gromov hyperbolicity, quasiconformal mappings etc in Euclidean spaces. Based on the theory of the coarse quasi-mappings, by constructing a new class of mappings, we shall investigate an intrinsic relationship of Gromov hyperbolic spaces, coarse quasi-mappings etc. ..(3) Characterization of quasi-isometric mappings with respect to quasihyperbolic metric in metric spaces: We shall establish a necessary and sufficient condition for a homeomorphism to be quasi-isometric with respect to quasihyperbolic metric in metric spaces. As an application of the obtained condition, we plan to study an related open problem raised by Vaisala in 1999...This study has very important theoretical significance.