Riemann-Hilbert problems, circle packing and conformal geometry

该项目的总体目标是研究保形几何和圆填充中的非线性边值问题。特别注意将针对应用几何函数理论和微分几何以及建设性的方法在圆包装。重点是系统地研究了Riemann-Hilbert-Poincaré问题（RHPPs），这些问题的特征在于包含解的值ω及其共形变形的边界条件|ω′|.这些所谓的共形RHPPs类包括所有的Riemann-Hilbert问题和共形几何中的许多自由边值问题，特别是Beurling型问题。在RHPPs的应用中，我们发现保形几何中的Berger-Nirenberg问题和Bergman空间中函数的零集分析。共形RHPPs具有典型的属性，使他们能够使用新开发的工具。特别是Beurling推广的Riemann映射定理、PDE方法和圆填充的新技术的最新发展，为共形RHPPs的研究提供了有效的途径。基于这些新的想法，我们将探讨不同方面的共形RHPPs：存在性，唯一性和性质的解决方案的经典问题和他们的离散对应，计算解决方案的数值算法，和收敛下细化的离散化。

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.