Geometry of Conformal and Quasiconformal Mappings

共形和拟共形映射的几何

• 批准号: 0103626
• 负责人: Christopher Bishop
• 金额: $ 17.64万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103626&HistoricalAwards=false
• 关键词: Geometry; Conformal; Quasiconformal; Mappings

Abstract for DMS - 0103626The PI, Christopher Bishop, will study the geometric propertiesof conformal mappings in the plane and quasiconformal mappings in space,focusing on the expansion properties of such maps and investigatingvarious applications to geometric function theory, dynamics andtopology. The PI has shown that a result of Dennis Sullivan'sconcerning the geometry of convexbodies in hyperbolic three space implies a factorizationtheorem for conformal mappings in the plane and this, in turn,implies uniform bounds on the amount ofcontraction a conformal map in the plane can have. Finding the best constantsin the factorization theorem has consequences for wellknown problems such as dimension distortion, integral means and Brennan'sconjecture. The PI will continue his work onlimit sets of Kleinian groups, a natural and important class of fractal sets.The questions here are mainly to estimate the fractal dimension of these setsand study the behavior of the dimension as the group is deformed. The PI will also work on the metric propertiesof harmonic measures, particularly results which quantify the idea that harmonicmeasure cannot be concentrated on a small set. Problems include thelower density conjecture, stability of harmonic measure and the growth rateof diffusion limited aggregation. A few other questionsinvolving quasiconformal and biLipschitz maps are also considered.Conformal mappings are a class of functions which are importantin many area of mathematics and which are closely relatedto mnay physical problems (fluid flow, heat conduction, electric fields, random growth models, ...) and have been intensively studied for many years. One of the fundamental properties of such maps is expansion; they tend to push points farther apart on average. Making this precise has motivated much research in mathematical analysis.The PI has discovered a new way of quantifyingthis expansion by approximating conformal maps by (the moregeneral class of) quasiconformal maps and showing theseapproximations may be taken with a very strong expansion property. This has given a clearer understanding of someknown results and has led to progress on new problems.In particular, it implies new results about Kleinian groups(these are important examples of conformal dynamical systems, and hence a contribution to the more general area of dynamical systems, fractals and chaos). The PI's approach also ties the behavior of conformal maps to the geometry three dimensional hyperbolic space; this connection seems to be new and should lead to many interesting problems and more interaction between the areas of complex analysis and three dimensional topology (already connected in other ways). He will also investigate the computationalaspects of this connection which may lead to new methods of computing conformal maps and Greens functions (importantfor a variety of applications). The PI will also continue his investigation of other problems including the geometry of randompaths such as Brownian motion, the stability underperturbation of certain dynamical systems and fundamentalgeometric properties of conformal and quasiconformal mappings.

摘要 DMS - 0103626 PI Christopher Bishop 将研究平面中的共形映射和空间中的拟共形映射的几何特性，重点关注此类映射的扩展特性，并研究在几何函数理论、动力学和拓扑中的各种应用。 PI 表明，丹尼斯·沙利文 (Dennis Sullivan) 关于双曲三空间中凸体几何形状的结果暗示了平面中共形映射的因式分解定理，而这又暗示了平面中共形映射可以具有的收缩量的统一界限。 寻找因式分解定理中的最佳常数对于维数畸变、积分均值和布伦南猜想等众所周知的问题具有重要意义。 PI 将继续他对克莱因群极限集的研究，克莱因群是一类自然而重要的分形集。这里的问题主要是估计这些集合的分形维数，并研究该维数在群变形时的行为。 PI 还将研究调和测量的度量属性，特别是量化调和测量不能集中在一个小集合上的想法的结果。 问题包括低密度猜想、调和测度的稳定性以及扩散限制聚集的增长率。 还考虑了涉及拟共形和biLipschitz映射的其他一些问题。共形映射是一类在许多数学领域中都很重要的函数，并且与许多物理问题（流体流动、热传导、电场、随机增长模型等）密切相关，并且已经被深入研究了很多年。 此类地图的基本属性之一是扩展性。他们倾向于将平均点拉得更远。 使其精确化激发了数学分析方面的大量研究。PI 发现了一种量化这种展开的新方法，通过（更一般的一类）拟共形映射来近似共形映射，并显示这些近似可以具有非常强的展开特性。 这使得人们对一些已知的结果有了更清晰的理解，并在新问题上取得了进展。特别是，它意味着关于克莱因群的新结果（这些是共形动力系统的重要例子，因此对动力系统、分形和混沌等更普遍的领域做出了贡献）。 PI 的方法还将共形映射的行为与几何三维双曲空间联系起来；这种联系似乎是新的，应该会导致许多有趣的问题以及复杂分析和三维拓扑领域之间的更多交互（已经以其他方式连接）。 他还将研究这种连接的计算方面，这可能会带来计算共形图和格林函数的新方法（对于各种应用都很重要）。 PI 还将继续研究其他问题，包括随机路径的几何形状（例如布朗运动）、某些动力系统的稳定性欠扰动以及共形和拟共形映射的基本几何特性。