Riemann-Hilbert problems, circle packing and conformal geometry

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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.