# Why there are two different area preserving mean curvature flow ?

I see two different area preserving mean curvature flow.

First, in 260th page of Flow by mean curvature of convex surfaces into spheres, $$\partial _t F = -H\nu + \frac{1}{n} h_1 F$$ where $F$ is position vector, $H$ is mean curvature, $\nu$ is normal vector, and $$h_1=\frac{\int H^2 d\mu}{\int d\mu}$$

Second, in Stability of the surface area preserving mean curvature flow in Euclidean space, the area preserving mean curvature flow is $$\partial_t F =(1-h_2H)\nu$$ where $h_2$ is $$h_2 = \frac{\int H d\mu}{\int H^2 d\mu}$$

I try to prove they are equal, but fail. Whether they are equal ? how to show it ?

• Your question is explicitly answered near the top of the second page of the second paper. – Anthony Carapetis Sep 14 '17 at 13:01
• @AnthonyCarapetis Thanks , I read the version of arxiv, there is a little difference. Could talk about normalized variant ? I don't know it . – lanse7pty Sep 14 '17 at 13:31

They are different flows. Huisken's is designed to produce the same surfaces as the normal mean curvature flow, just rescaled to have constant area, for the purposes of analysing the singularity of MCF. The one considered in the second paper is the same as that studied by McCoy, and gives rise to genuinely different geometry (unless your initial surface is very symmetric).

One of the most obvious differences is that Huisken's equation depends on the choice of origin, while the other is translation-invariant.