2020/09/22 by Alencar, Hilário, Neto, Gregório Silva, Zhou, Detang
#35K10 #35K15 #53E44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Primary 53C42 #Secondary 53E10
paper · doi:10.48550/arxiv.2009.10494
In this paper we prove rigidity results for two-dimensional, closed, immersed, non-necessarily convex, self-similar solutions of a wide class of fully non-linear parabolic flows in ℝ3. We show this self-similar solutions are the round spheres centered at the origin provided it has genus zero and satisfies a suitable upper pinching estimate for the Gaussian curvature. As applications, we obtain rigidity results for the round sphere as the only closed, immersed, genus zero, self-similar solution of several well known flows, as the flow of the powers of mean curvature, the harmonic mean curvature flow and the α-Gaussian curvature flow for α∈(0,1/4). We remark that our result does not assume any embeddedness condition.