Geometric Analysis and Topology Seminar
Mean Curvature Flow of Reifenberg Sets
Speaker: Or Hershkovits, NYU
Location: Warren Weaver Hall 517
Date: Friday, October 30, 2015, 11 a.m.
Synopsis:
The mean curvature flow, the gradient flow of the area functional, is one of the most natural geometric flows to consider for embedded hypersurfaces in \(R^{n+1}\). Classically, given a sufficiently smooth hypersurface (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 hypersurface 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 hypersurfaces (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.
In 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 \((\varepsilon,R)\) Reifenberg sets have a weak metric notion of a tangent hyperplane at every point and scale \(r<R\) (with accuracy determined by \(\varepsilon\)), but those tangents are allowed to tilt as the scales vary.
We show that if \(X\) is an \((\varepsilon,R)\) Reifenberg set with \(\varepsilon\) 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.