Given a smooth compactly embedded initial hypersurface $M_0$ and let $M_t$ ($0<t<T$) be a smooth solution of mean curvature flow starting from $M_0$. Then how does the regularity of $M_t$ depend on $M$? I'm particularly interested in the dependence of $C^{1,\alpha}$ regularity of $M_t$ on $M_0$. Does the $C^{1,\alpha}$ norm of $M_t$ depend only on $T$ and the $C^{1,\alpha}$ norm of $M_0$?
This is just quite a naive question, and I believe there is an affirmative answer. For example, in the paper "higher regularity of the inverse mean curvature flow" by Huisken and Ilmanen, https://projecteuclid.org/euclid.jdg/1226090483, the authors have used this result in the proof of Lemma 2.6. However I just don't know where I can find a proof of this result.
Can anyone give me a reference where a rigorous statement and proof is given? I'll really appreciate your help!
