Self - shrinkers of Mean Curvature Flow and Harmonic Map Heat Flow with Rough Boundary Data

In this thesis, first, joint with Longzhi Lin, we establish estimates for the harmonic map heat flow from the unit circle into a closed manifold, and use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic. Second, we prove the uniqueness for energy decreasing weak solutions of the harmonic map heat flow from the unit open disk into a closed manifold, given any H' initial data and boundary data, which is the restriction of the initial data on the boundary of the disk. Previously, under an additional assumption on boundary regularity, this uniqueness result was obtained by Riviere (when the target manifold is the round sphere and the energy of initial data is small) and Freire (for general target manifolds). The point of our uniqueness result is that no boundary regularity assumption is needed. Also, we prove the exponential convergence of the harmonic map heat flow, assuming that the energy is small at all times. Third, we prove that smooth self-shrinkers in the Euclidean space, that are entire graphs, are hyperplanes. This generalizes an earlier result by Ecker and Huisken: no polynomial growth assumption at infinity is needed. Thesis Supervisor: Tobias H. Colding Title: Levinson Professor of Mathematics