Mean Curvature Flow: Smoothing, Regularity and Isoperimetric Properties

The key purpose of this work is to describe various properties of flow by mean curvature of embedded sub-manifolds. More precisely we will address questions relating to short time smoothing of the initial data via the flow, long time regularity properties of the flow and isoperimetric properties of the flow. The smoothing results apply to a class of objects called epsilon R Reifenberg sets, for which we show short time existence and uniqueness of smooth flow. This class is general enough to allow fractional Hausdorff dimension, and is, at least qualitatively, the roughest class for which a smooth flow is known to exist in arbitrary dimension an codimension. The regularity results, which are joint with Robert Haslhofer, extend Brian White's structure theory of mean convex flows to the setting of arbitrary ambient manifolds, removing a stumbling block that was left after White's work. The last results describe how singular, arbitrary codimensional mean curvature flow can be used to provide good fillings to cycles in Euclidean space. This leads to a very short proof for an isoperimetric inequality with a very good (although not optimal) constant. Through an investigation of the geometric measure theory of parabolic space time, culminating with a co-area formula in that context, a bound on the space-time measure of the flow is also obtained.