Closed mean curvature flows with prescribed tangent flows

Abstract

Given an embedded shrinker $\hat{I}\pounds$ in ${\mathbb{R}}^{n+1}$ that is either closed, asymptotically conical, or a Cartesian product of such a shrinker with ${\mathbb{R}}^k$, we construct a closed embedded mean curvature flow whose tangent flow at the first singularity is modeled on $\hat{I}\pounds$. We also prescribe the first-order asymptotics of the tangent flow. This result is a consequence of a more general theorem that allows us to construct mean curvature flows with an additional force whose tangent flow and first-order asymptotics at the first singularity are prescribed.

40pages, comments are welcome!