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 present some numerical evidence for a further new family of noncompact self-shrinkers with odd genus and two asymptotically conical ends. This is joint work with Huy Nguyen and Mario Schulz.
Geometric Analysis Seminar: Noncompact self-shrinkers for mean curvature flow with arbitrary genus
Speaker: Reto Buzano from the Università di Torino