Ancient Solutions to Geometric Flows

Ancient solutions to geometric flows - Panagiota Daskalopoulos

Panagiota Daskalopoulos
Columbia University
May 20, 2019
Abstract: Some of the most important problems in geometric evolution partial differential equations are related to the understanding of singularities. This usually happens through a blow up procedure near the 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 −∞ T≤+∞. We refer to them as ancient solutions. The classification of such solutions often sheds new insight upon the singularity analysis.

In this lecture series we will discuss Uniqueness Theorems for ancient solutions to parabolic partial differential equations, starting from the Heat equation and extending to the Semi-linear heat equation, the Mean curvature flow, the Ricci flow and the Yamabe flow. We will also discuss the construction of new solutions from the gluing of two more solitons.