On the Kaehler-Ricci Flow and Related Problems

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ABSTRACT DMS - 0206847.Abstract of the ProjectThis project is centered around the study of the Ricci flow on complete,non-compact Kaehler manifolds of nonnegative holomorphic bisectionalcurvature. The main goal is to seek a complete understanding of singularbehavior of the maximal solutions to the Kaehler-Ricci flow withnonnegative curvature, in particular the geometry, such as the rates of curvature decay and volume growth of geodesic balls, of complete Kaehler-Ricci solitons,which turn out to be the models for Type II singularities and can be considered as anatural extension of Calabi-Yau metrics on non-compact complexmanifolds. Progress on the project will lead to new understanding of geometry, analysis, and complex structure and could have important application to solving awell-known conjecture in complex geometry. The research will be basedon the effort of extending the dimension reduction method of Hamiltonfor the Ricci flow in the Riemannian case to the Kaehler case where onehas weaker curvature assumption. The proposed project deals with singular behavior of the Kaehler-Ricciflow, or Parabolic-Einstein equations, and studies uniformization typeproblems for complete Kaehler manifolds. The Kaehler-Ricci flow is animportant type of geometric evolution equations, which have profoundimportance and applications in science and geometry. Some of the examples 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, thereproduction of sparse species, and image sharpening. The knowledgegained in this project would not only have significant implications toour understanding of complex geometry which in turn could be very usefulto the study of mathematical physics, but also may lead to newunderstanding of singularity formations in other geometric evolutions.------------------------------------------------------------------------