Principal Investigator: Huai-Dong Cao
The Ricci flow, introduced by Richard Hamilton, has become one of the most powerful tools in geometric analysis. In the past twenty years or so, Hamilton has proved many remarkable theorems in the Ricci flow and developed a remarkable program to approach the Poincare conjecture and Thurston's geometrization conjecture using the Ricci flow. More recently, Perelman has made astounding breakthrough in the Ricci flow with the proof of a local injectivity radius estimate valid for all dimensions and used it to study the geometrization of three-manifolds. In addition to the applications of the Ricci flow to three-manifolds, many exciting possibilities remain. In this proposal, we propose to investigate several important problems in the Ricci flow and the Kaehler-Ricci flow which are of great interest in geometry, topology, nonlinear partial differential equations and complex analysis. They include studying stability/instability of Einstein metrics of positive scalar curvature (and more generally of shrinking Ricci solitons), constructing new Ricci solitons, seeking new Einstein metrics via the Ricci flow, aspects of geometrization of 4-manifolds, studying the asymptotic behavior of solutions to the Kaehler-Ricci flow on compact Kaehler manifolds with positive first Chern class, and the uniformization of complete noncompact Kaehler manifolds of positive curvature.
The Ricci flow is an important type of geometric flows (or geometric evolution equations) which have profound importance and applications in science and geometry. Examples of applications of other include the motion of a surface by its mean curvature, the flow of gas in a porous mechanism, the motion of a liquid crystal, the diffusion of oil in shale, the reproduction of sparse species, and image sharpening.
