Calculus of Variations and Geometric Measure Theory

R. Buzano - H. T. Nguyen - M. B. Schulz

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

created by muller on 13 Jun 2022


Submitted Paper

Inserted: 13 jun 2022
Last Updated: 13 jun 2022

Year: 2021

ArXiv: 2110.06027 PDF


In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.