Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions

The goal of this paper is to relax convexity assumption on some classical results in mean curvature flow. In the first half of the paper, we prove a generalized version of Hamilton’s differential Harnack inequality which holds for mean convex solutions to mean curvature flow with a lower bound on $\frac{{\lambda }_1}{H}$ where ${\lambda }_1$ is the smallest principal curvature. Then, we use classical maximum principle to provide several characterizations of family of shrinking spheres for closed, mean convex ancient solution to mean curvature flow with a lower bound on $\frac{{\lambda }_1+\mathrm{.}\mathrm{.}+{\lambda }_k}{H}$ for some $1\le k\le d-1$, where ${\lambda }_1\le {\lambda }_2\le \mathrm{.}\mathrm{.}\le {\lambda }_d$ are the principal curvatures.