Title: Mean Curvature Flow of Reifenberg Sets.
Speaker: Or Hershkovits
Speaker Info: Courant Institute
Brief Description:
Special Note:
Abstract:
The mean curvature flow, the gradient flow of the area functional, is one of the most natural geometric flows to consider for embedded hyper-surfaces in R^{n+1}. Classically, given a sufficiently smooth hyper-surface (for which both the area and its gradient are defined), there exists a unique flow starting from it that exists for some positive time. Moreover, the flow smooths the hyper-surface instantaneously. In the early 90s it was shown by Ecker and Huisken that the smoothness assumption can be weakened to the class of uniformly locally Lipschitz hyper-surfaces (for which the area is defined, but its gradient may not be). When n>1, this is the least regular object for which the flow was known to exist.Date: Monday, December 7, 2015In this talk, we will discuss the short time existence and uniqueness of smooth mean curvature flow in arbitrary dimension starting from a class of sets which is general enough to include some fractal sets (for which even the area is not defined).Those so-called (e,R) Reifenberg sets have a weak metric notion of a tangent hyper-plane at every point and scale r
We show that if X is an (e,R) Reifenberg set with e sufficiently small, there exists a unique smooth mean curvature flow emanating from X. When n>1, this provides the first known example of instant smoothing, by mean curvature flow, of sets with Hausdorff dimension larger than n. If time permits, we will discuss the arbitrary co-dimensional case.