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.