The Regularity of Cauchy-Riemann Mappings and Solutions of Systems of Nonlinear Partial Differential Equations

本研究项目的目的是研究出现在几个复杂变量的偏微分方程组的解。本研究的重要数学应用包括建立解的正则性和只知道泰勒级数系数的解的收敛性。物理系统的重要应用包括统计力学和气象学中出现的方程。例如，平面二聚体模型是空间中某些二维表面的统计力学模型，它是整数上简单随机游走的高维推广。这种曲面的极限形状是用一个与本项目研究的复Burgers方程密切相关的偏微分方程来建模的。同样的方程也适用于研究模拟某些大气现象行为的准地转方程。该项目的主要部分涉及理解几何条件的问题，这些几何条件意味着两个柯西-黎曼（CR）流形之间的足够光滑的CR映射在流形光滑时是光滑的，当流形是实解析时是实解析的。与此密切相关的问题包括抽象CR流形上CR函数的局部正则性和微局部正则性。可以使用的工具包括一个新的傅立叶-兄弟- iagolnitzer变换族，解析盘，以及在超曲面上CR函数的全纯可扩展性理论中发展的方法。

The aim of this research project is to study the solutions of systems of partial differential equations that arise in several complex variables. Important mathematical applications of this research include establishing the regularity of solutions and the convergence of solutions for which only Taylor series coefficients are known. Important applications to physical systems include equations that arise in statistical mechanics and meteorology. For example, the planar dimer model is a statistical mechanical model of certain two-dimensional surfaces in space that is a higher dimensional generalization of the simple random walk on integers. The limit shape of such surfaces is modeled by a partial differential equation closely related to the complex Burgers' equation studied in this project. The same equation is also relevant in the study of the quasi-geostrophic equations that model the behavior of certain atmospheric phenomena. The main part of the project involves the problem of understanding geometric conditions that imply that a sufficiently smooth CR mapping between two Cauchy-Riemann (CR) manifolds is smooth when the manifolds are smooth, and real analytic when the manifolds are real analytic. Closely related problems that will be investigated include the local and microlocal regularity of CR functions on abstract CR manifolds. The tools that may be used include a new family of Fourier-Bros-Iagolnitzer transforms, analytic discs, and the methods developed in the theory of holomorphic extendability of CR functions on hypersurfaces.