双曲型度量几何与映射的连续模

Hyperbolic-type metrics have found numerous applications in the quasiconformal mapping theory and related areas. However, the study of hyperbolic-type geometry itself is few and the precise analysis on the geometry has turned out to be quite challenging. The geometry of hyperbolic-type metrics is one of the main topics of this research program, and geometric and analytic properties of quasiconformal mappings are also going to be studied by use of geometry of hyperbolic-type metrics. Firstly, the uniqueness of the quasihyperbolic geodesics will be investigated, and then the convexity and smoothness of quasihyperbolic balls will be established. Secondly, the ubiquity of so-called ‘near-geodesics’ of the hyperbolic-type metrics and the convexity of hyperbolic-type balls are going to be studied. Then the properties of maps between hyperbolic-type metric spaces, including the modulus of continuity of quasiconformal mappings with respect to hyperbolic-type metrics, are about to be studied. Finally, we will characterize the quasiconformality of maps with some modulus of continuity with respect to hyperbolic-type metrics and also give some asymptotically sharp estimates of the coefficient of quasiconformality. Some Schwarz-type lemmas with respect to hyperbolic-type metrics will be also established in this project. The study of this project will help us to understand the geometry of quasiconformal mappings and their generalizations to metric spaces, and to provide useful tools for the analysis on metric measure spaces.

双曲型度量已经在拟共形映射理论和相关的领域中有着非常重要的应用，然而对双曲型度量几何本身的研究却很少，对其精细的分析具有很大的挑战性。本项目主要研究双曲型度量几何，并利用区域的双曲型度量几何来研究拟共形映射的几何和分析性质。首先，着重研究拟双曲度量测地线的唯一性，并且利用所得的结果来研究拟双曲度量球的凸性和光滑性。其次，研究双曲型度量空间的近测地线的万有性和双曲型度量球的凸性。然后，利用双曲型度量几何来研究双曲型度量空间之间映射的性质和拟共形映射在双曲型度量下的连续模。最后，我们将刻画具有在双曲型度量下的某种连续模的映射的拟共形性，并且给出拟共形性系数的渐进精确的估计，建立拟共形映射的关于某些双曲型度量的Schwarz型引理。本项目的研究可以使人们对拟共形映射的几何及其在度量空间中的推广有更好的认识，为拟共形映射理论在度量测度空间上的分析理论提供更多的应用工具。

双曲型度量作为双曲度量的推广在几何函数论中有着重要的应用。本项目对双曲型度量几何本身进行了系统的研究，并将所得结果应用于双曲度量空间之间的映射的拟共形性和连续模的研究。通过研究我们获得了几类双曲型度量空间中的近测地线的球凸性质，利用球凸性质证明了双曲型度量球的局部凸性；利用拟双曲度量权函数满足的 Dini 类型的条件给出了一致凸 Banach 空间中的拟双曲测地线的光滑性的判别准则；证明了一致光滑 Banach 空间中拟双曲球的光滑性以及距离比度量空间中球的星形性；证明了一个 Gromov 双曲度量和经典双曲度量之间的精确比较关系，从而解决了一个相应的公开问题；给出了双曲型度量球之间、双曲型度量球和欧氏球之间的球包含关系；刻画双曲型度量空间之间的Lipschitz映射的拟共形性，并给出拟共形性常数的渐进精确的估计；给出了双曲 Lambert 四边形一对邻边的双曲长度的和以及乘积的精确上下界，揭示了反双曲正弦函数的广义凹凸性；获得了雅克比椭圆函数的Holder凸性以及在矩形区域中的双曲度量密度的估计；证明了广义椭圆积分的凹凸性、不等式、初等逼近等；发现了零平衡超几何函数和广义模函数的变换不等式和无穷乘积公式等。本项目对双曲型度量空间的几何性质做了比较完善的研究，将双曲几何的一些良好性质在一般的空间区域上作了推广和深化。这些结果将为高维几何函数论的研究提供了一些重要的基础和工具。

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