Contraction of surfaces by harmonic mean curvature flows and nonuniqueness of their self similar solutions

Abstract.

We consider the evolution equations $F_t=-(H_{-1})^{\alpha}\nu$, where $0<\alpha<1$, $\nu$ is the unit outer normal vector and $H_{-1}$ is the harmonic mean curvature defined by $H_{-1}=((\kappa_1^{-1}+\kappa_2^{-1})/2)^{-1}$. In this paper, we prove the nonuniqueness of their strictly convex self similar solutions for some $0<\alpha<1$. This result implies that there are non-spherical self similar solutions.