Linear and Nonlinear Analysis on Complete Kahler Manifolds

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ABSTRACT DMS - 0203023.PI: Lei NiThe principle investigator proposes to study the interplay between thegeometry and the analysis on complete Kaehler manifolds. The focus will belinear and nonlinear analysis on such manifolds. The main tool is solvingthe linear equation, such as the Poisson equation and Poincare-Lelongequation, and the nonlinear equations such as the Kaehle-Ricci flow. Thegoal is to understand the space of holomorphic functions (plurisubharmonicfunctions), the interplay between the geometry and the function theory andapplying the results to the uniformization of complete Kaehler manifoldswith nonnegative curvature.The manifold is the space where every physical event happens. Theglobal analysis on manifolds studies the overall properties of themanifolds by piecing together the local information. Kaehler manifolds arethe basic block in the universe model according to the string theory. The proposed studyhas close connection with the theory of general relativity and stringtheory. The nonlinear differential equations studied in the proposal haveapplications in the study of the structure of complicated molecules,liquid-gas boundary, and even the large scale networks.