Geometric Analysis Seminar
We show that after a perturbation on the initial data of mean curvature flow, the perturbed flow can avoid certain non-generic singularities. This contributes to the program of dynamical approach to mean curvature flow initiated by Colding and Minicozzi. The key is to prove that a positive perturbation on initial data would drift to the first eigenfunction direction after a long time. This result can be viewed as a global unstable manifold theorem in the most unstable direction for a nonlinear heat equation. This is joint work with Jinxin Xue.