We will talk about properties of geometric evolution flows. The emphasis will be on the mean curvature flow. We will discuss basic properties of the flow, such as the evolution equation of geometric quantities, characterization of a singular time. We will also derive Huisken's monotonicity formula that will be used in getting self-shrinkers as the singularities of the flow. We will discuss properties of self-shrinkers. Interior curvature estimates derived by localization arguments will be also discussed. We will introduce Gaussian density and how Gaussian density can be used in singularity analysis. In the context of Gaussian density we will prove famous gap theorem of B. White.
Regularity of mean curvature flow by Klaus Ecker