Ricci Flows and Steady Ricci Solitons

This award supports research in differential geometry focusing on Ricci flows. These flows are defined on manifolds equipped with a metric, that is to say, a way of measuring distance. A Ricci flow is a geometric partial differential equation for Riemannian metrics. The Ricci flow tends to evolve an initial metric into a more homogeneous one. The singularity analysis of Ricci flow is a central subject, as it helps to understand the geometry and topology of manifolds. The most remarkable application in this direction is the resolution of the Poincare conjecture and the Geometrization conjecture by Perelman. Many of the Ricci flow singularity models are Ricci solitons. Recent examples of solitons constructed by the PI look like flying wings. The PI will study the geometry of all 3-dimensional steady Ricci solitons and try to classify them by their asymptotic limits. In addition, the PI will study the higher-dimensional steady Ricci solitons and see if they can arise as singularity models. The research project is split into two projects. The first project is to prove the O(2)-symmetry of all 3-dimensional steady Ricci solitons. This includes showing that the Bryant soliton is the unique 3-dimensional steady Ricci soliton that is asymptotic to a ray. This extends a previous result of the PI in which one assumes the O(2)-symmetry of the soliton. The PI developed some methods that may be extended to the more general class of ancient collapsed Ricci flows in dimension 3. In particular, the PI will investigate the symmetry of the ancient collapsed Ricci flows in dimension 3 and aim at classifying them by certain 2-dimensional limits. The second project is a continuation of the PI's work on the existence theory of Ricci flows coming out of non-compact initial manifolds. The PI will investigate the applications of non-compact Ricci flows in topology and geometry. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.