May 26, 2021
TITLE: Ancient compact solutions to Ricci flow and Mean curvature flow
Some of the most important problems in partial differential equations are related to the understanding of singularities. This usually happens through a blow up procedure near the potential singularity which uses the scaling properties of the equation. In the case of a parabolic equation the blow up analysis often leads to special solutions which are defined for all time , for some . We refer to them as ancient solutions. The classification of such solutions often sheds new insight to the singularity analysis.
In this lecture we will discuss Uniqueness Theorems for ancient compact solutions to the Ricci flow and Mean curvature flow. Emphasis will be given to the complete classification of compact κ-noncollapsed solutions to the 3- dim Ricci flow. We will also discuss the 2-dim case where the κ-noncollapsed condition is not necessary for uniqueness.