Geometry and classifications of Ricci solutions

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It is a central theme in mathematics to categorize and classify geometric objects. One effective way of doing so is to prescribe a canonical metric for each object, namely a canonical way to measure the distance in an object. It is a classical result that each two dimensional surface carries a certain metric with constant curvature. For the three dimensional case, the problem becomes vastly more difficult. It was a much celebrated achievement of Perelman who successfully tackled the problem about ten years ago. He adopted an approach developed by Hamilton who has devised a way of deforming an arbitrary metric via a set of differential equations. A crucial step in understanding this set of equations for Perelman is to classify the so-called self-similar solutions. The current project aims to extend Perelman's work to dimension four and beyond. It is expected that the results will aid in the study of four dimensional spaces. Ricci solitons, as self-similar solutions to the Ricci flows, play a central role in the singularity analysis of the Ricci flows. The main theme of this project is to study shrinking Ricci solitons. In dimensions two and three, they have been completely classified. The classification has found important applications in the resolution of the Poincare and geometrization conjecture for three dimensional manifolds. The current project aims to obtain a classification for four dimensional complete gradient shrinking Ricci solitons, building upon the recent progress including curvature estimates and a description of the structure at infinity. Another goal of the project is to lay some foundation toward a possible classification for high dimensional gradient shrinking Ricci solitons.