Linear and Nonlinear Analysis on Complete Kahler Manifolds

University Of California-San Diego, La Jolla CA

Investigators

Abstract

ABSTRACT DMS - 0203023. PI: Lei Ni The principle investigator proposes to study the interplay between the geometry and the analysis on complete Kaehler manifolds. The focus will be linear and nonlinear analysis on such manifolds. The main tool is solving the linear equation, such as the Poisson equation and Poincare-Lelong equation, and the nonlinear equations such as the Kaehle-Ricci flow. The goal is to understand the space of holomorphic functions (plurisubharmonic functions), the interplay between the geometry and the function theory and applying the results to the uniformization of complete Kaehler manifolds with nonnegative curvature. The manifold is the space where every physical event happens. The global analysis on manifolds studies the overall properties of the manifolds by piecing together the local information. Kaehler manifolds are the basic block in the universe model according to the string theory. The proposed study has close connection with the theory of general relativity and string theory. The nonlinear differential equations studied in the proposal have applications in the study of the structure of complicated molecules, liquid-gas boundary, and even the large scale networks.