Principal Investigator: Lei Ni
The principle investigator proposes to study the interplay between the geometry and the analysis on complete Kaehler manifolds. The focus will be the linear and nonlinear parabolic equations on complete manifolds with various curvature assumptions. These equations includes the linear heat equation and Laplacian operator, harmonic mapping heat equation, mean curvature flow, Hermitian-Einstein flow, Ricci/Kaehler-Ricci flow, etc. These equations have various physical and geometric origins. But they share many common features such as geometric symmetries and identities, monotonicity formulae, entropy like considerations, differential Harnack (also called Li-Yau-Hamilton) inequalities. They also connect to each other in various ways. By studying them together, more light is shed on all of them.
Differential Geometry is the study of the relationship between the geometry of a space, a manifold in the mathematical notion, and the analytic properties of the functions and the differential equations, on the underlying space. Geometric analysis is the study of the overall geometric and topological properties of a space by piecing together the local information. Since the spaces are usually curved ones, the ``curvature" was introduced to measure the deviation from the Euclidean space and the techniques are often `nonlinear' even dealing with a linear problem. The study of this area of mathematics has close connection with the general relativity and string theory in physics. The applications can be found in the study of thestructure of complicated molecules, liquid-gas boundaries, and even the large scale networks. This project on studying the Kaehler manifolds via linear and nonlinear parabolic equations will enhance the understanding of geometric analysis, linear and nonlinear partial differential equations, algebraic geometry and mathematical physics.
