Natasa Sesum

### Subtitle:

Mean curvature flow

### Text:

### Prerequisites:

16:640:517 - Partial Differential Equations I

### Description:

We will introduce the mean curvature flow, which is one of well studied geometric flows in which the hypersurface in R^{n+1} evolves in the normal direction by the speed given by its mean curvature.

We will go over Husiken's paper in which he introduces the mean curvature flow and shows that every closed convex hypersurface evolving by the MCF must shrink to a point in finite time, in a spherical manner. We will introduce Huisken's monotonicity formula and talk about how it can be used to study singularities that are inevitable in the case of a closed MCF. We will also talk about classification of singularities occurring in the MCF, which is possible if extra conditions on the hypersurface are being assumed (for example, in the case we start the flow from a mean convex closed hypersurface, the property that turns out to be preserved along the flow).

We will also talk about the mean curvature flow starting with an entire graph, an example of a complete, mean curvature flow. In this case we will learn a few useful tricks, how to make a use of cut off functions to apply maximum principle and get useful curvature interior estimates. As a consequence we will see that the MCF starting at an entire graph exists forever.

If time permits we will also discuss ancient solutions to the mean curvature flow (solutions that exist from time equal to -infty and that occur as singularity models) and their classification in certain cases.