Caltech/UCLA Joint Analysis Seminar

For any integer k>1 there exist smooth solutions Mt (t<0) of MCF that form a one-point singularity at time t=0, after which there exist at least 2k forward evolutions Mt1, ..., Mtk, Nt1, ... , Ntk (t>0) by the flow. The solutions Mtj and Ntj are topologically distinct. The analogous statement for Ricci Flow also holds, and I will explain both.

Building on these self similar solutions to MCF, I will also describe non-self similar solutions that have a given cone as their initial data. One conclusion is that for any k>1 there is a smooth self similar solution to MCF that forms a one point singularity, and for which the set of possible smooth forward evolutions contains a k-dimensional continuum. Another conclusion is that the set of smooth solutions to MCF whose initial condition is one of the stationary cones in ℝn (n∈{4, 5, 6, 7}) is infinite dimensional .