Singular Integrals in Geometric Analysis

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Geometric Analysis (GA) is a branch of mathematics treating the interface of Differential Geometry (DG) and Partial Differential Equations (PDE). This field includes both the use of PDE-methods in DG and the motivation of DG-results in PDE. In the study of GA, it is naturally and frequently needed to find complete integrability conditions for the existence of metrics with certain curvature conditions. Such a finding often involves various kinds of singular integrals that exist more or less in Harmonic Analysis (HA). The proposed research program will shed light upon this issue through the following three p-Laplace oriented objectives: 1. p-Green's functions and their geometric properties; 2. Minkowski's problem for generalized torsional rigidity; 3. Morrey-Riesz's integrals with applications to some geometric PDE. The novelty of this research program is to utilize new HA tools and techniques (especially, an effective incorporation of Riesz potentials and Morrey capacities plus Choquet integrals) to settle some fundamental problems on singular integrals arising from GA. This research program, creating a synthesis among a host of distinct topics, will not only be useful to researchers in both analysis and geometry based on either capacity or curvature, but also be of interest to both physicists and engineers in certain areas.