Picture below is from 258th page of Huisken, Gerhard, Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom. 20, 237-266 (1984). ZBL0556.53001.
I want to get the red line, assume $\gamma$ is geodesic starting at $x$ with arc length parameter $\gamma$, then $$ \nabla_{\dot \gamma} H=\lim_{\gamma\rightarrow 0} \frac{H(\gamma)-H_{\max}}{\gamma} $$ And $$ |\nabla_{\dot\gamma}H| \le |\nabla H| \le \eta^2 H_\max^2 $$ let $\gamma = \eta^{-1}H_\max^{-1}$, then $$ H_\max-H(\gamma) \le \eta H_\max $$ so, $$ H\ge(1-\eta)H_\max $$ but for other points ,$\gamma < \eta^{-1}H_\max^{-1}$, I can't get it. How to show it? Besides, I think the above process has some wrong.
