共形不变量和拟共形映射及其应用

拟共形映射理论的中心问题之一是研究各种共形不变量和共形不变度量在拟共形映射下的偏差和形变。本项目研究共形不变量和共形不变度量的几何性质，并利用它们来刻画拟共形映射的几何和分析性质。我们的主要目标是发展拟共形映射的渐进精确的定量的偏差理论。本项目致力于得到一类极值共形模的显式的精确的上下界，并获得精确的拟共形映射的偏差定理；给出关于共形模度量及其对偶度量的等距映射的本质特征的刻画，进而证明关于共形模度量及其对偶度量的等距映射的共形性猜想。作为拟共形映射的精确偏差定理的应用，我们将研究拟共形齐次区域的刻画，最终目标是解决Bonfert-Taylor、Canary和Taylor等人提出的寻找拟共形齐次常数的显式的下界的公开问题和证明Gehring关于亏格0的双曲曲面的拟共形齐次性的猜想。通过本项目的研究，预期改进和扩展一些经典定理，解决经典拟共形映射理论中的一些深入而困难的问题。

One of the central problems in the quasiconformal mappings theory is to study the deformation of various conformal invariants and conformally invariant metrics under these mappings. The topics of the proposed research are to investigate the geometric properties of conformal invariants and conformally invariant metrics, and to use conformal invariants and conformally invariant metrics to study the geometric and analytic properties of quasiconformal mappings. During the course of the project the aim is to develop a quantitative sharp distortion theory of quasiconformal mappings. We aim to obtain some explicit sharp bounds for extremal conformal module and provide sharp distortion theorem for quasiconformal mappings. We also plan to characterize the isometry with respect to the modulus metric or its dual metric, and to prove a conjecture on isometries of the modulus metric and its dual metric. As an application of the sharp distortion theorems of quasiconformal mappings, we will study the characterizations of quasiconformal homogeneity. Our main aim here is to find an explicit lower bound for the quasiconformal homogeneity constant, and to resolve the conjecture on the quasiconformal homogeneity of hyperbolic surface. All of these desired results will extend and improve some classical theorems in quasiconformal mappings theory.