Riemann-Hilbert problems, circle packing and conformal geometry

该项目的总体目标是研究保角几何和圆填充中的非线性边值问题。将特别关注几何函数论和微分几何中的应用，以及圆填充中的构造方法。重点是系统地研究黎曼-希尔伯特-庞加莱问题(RHPP)，该问题的边界条件涉及解的值ω及其共形失真|ω‘|。这类所谓的共形RHPP包括所有的Riemann-Hilbert问题和保形几何中的许多自由边值问题，特别是Beurling类问题。在RHPP的应用中，我们发现了共形几何中的Berger-Nirenberg问题和Bergman空间中函数的零集分析。共形RHPP具有典型的性质，这使得新开发的工具可以访问它们。特别是Beurling推广的Riemann映射定理、偏微分方程方法和圆填充新技术的发展，为研究共形RHPP提供了有效的途径。基于这些新思想，我们将探讨共形RHPP的不同方面：经典问题及其离散问题解的存在性、唯一性和性质，计算解的数值算法，以及在离散化精化下的收敛。

The general goal of the project is the study of nonlinear boundary value problems in conformal geometry and circle packing. Special attention will be directed to applications in geometric function theory and differential geometry as well as to constructive methods in circle packing. The focus is on a systematic study of Riemann–Hilbert–Poincaré problems (RHPPs) which are characterized by boundary conditions involving the values of the solutions ω and their conformal distortion |ω′|. The class of these so–called conformal RHPPs comprises all Riemann–Hilbert problems and many free boundary value problems in conformal geometry, in particular Beurling–type problems. Among the applications of RHPPs we find the Berger–Nirenberg problem in conformal geometry and the analysis of zero sets of functions in Bergman spaces. Conformal RHPPs have typical properties which make them accessible to newly developed tools. In particular, recent developments on Beurling’s extended Riemann mapping theorem, PDE methods and new techniques in circle packing provide effective approaches for studying conformal RHPPs. Based on these new ideas we will explore different aspects of conformal RHPPs: existence, uniqueness and properties of solutions of the classical problems and their discrete counterparts, numerical algorithms for computing solutions, and convergence under refinement of the discretization.