拟共形映射和Gromov双曲空间理论在分形几何、双曲群中的Cannon猜测和Kapovich-Kleiner猜测、Lie群的拟等距刚性问题等领域的研究中具有重要的作用。本项目主要研究Bonk,Heinonen和Koskela建立的拟共形一致化Gromov双曲空间理论在John度量空间中的一些应用，将拟共形映射与Gromov双曲空间理论结合起来，从Gromov意义下的负曲率性理解几何函数论，为相关问题的研究提供一种新的思路。主要研究内容如下:借助拟共形一致化Gromov双曲空间理论，我们首先讨论Gromov双曲John空间中拟双曲测地线的几何性质，探索欧氏空间中与维数无关的结论，并推广到一般度量空间。其次，我们计划利用共形模等工具研究Gromov双曲John度量空间与一致度量空间的关系。最后，我们将讨论John度量空间与一致度量空间上的拟共形映射、拟对称映射以及拟Mobius映射之间的关系。

The theory of quasiconformal mapping and Gromov hyperbolic spaces plays an important role in different fields, such as fractal geometry, Cannon conjecture and Kapovich-Kleiner conjecture in hyperbolic groups, quasi-isometric rigidity of Lie groups. The main aim of this project is to study the applications of quasiconformal uniformizing Gromov hyperbolic spaces theory found by Bonk,Heinonen and Koskela on John metric spaces. Combining quasiconformal mapping theory with Gromov hyperbolic spaces theory, we shall explain some results concerning geometric function theory from the negative curvature in the sense of Gromov and provide a new method for studying related questions. The main content of this project is the following: We shall first apply quasiconformal uniformizing Gromov hyperbolic spaces theory to discuss the geometric properties of quasihyperbolic geodesics on Gromov hyperbolic John spaces，we aim to search for dimension-free results on Euclidean spaces and generalize to general metric spaces. Then we will study the relationship between Gromov hyperbolic John spaces and uniform spaces by virtue of conformal modulus. Finally, we shall investigate the relationship between quasiconformal, quasimobius and quasisymmetric maps on John spaces and uniform spaces.