I want to know if the mean curvature flow has such a relationship ,and if so, what is like?
The solution $F_t$ to the Gaussian curvature flow $\partial_t{F}=-K\nu$ is the self-similar if for some constant $c(t)$ depending only on time, $\nabla (\partial_\tau{F})=c(t)\nabla {F}$,there $\partial_\tau{F}=-K\nu+X$ and X is tangential vector field.
So I want to know if there is a similar relationship for the mean curvature flow equation. Thanks a lot of!
There have another question. If $\nabla$ is the covariant derivative, then why is $$\partial_t \nabla_i \nabla_j=-\partial_t \Gamma_{ij}^{k} \nabla_k?$$