Work

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

Year: 2024

Type: article

Abstract: 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.

Source: Journal für die reine und angewandte Mathematik (Crelles Journal)

DOI: 10.1515/crelle-2024-0073

Authors: Reto Buzano, Huy The Nguyen, Mario B. Schulz

Institutions: University of Turin, Queen Mary University of London, University of Münster

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Deutsche Forschungsgemeinschaft, Engineering and Physical Sciences Research Council

Awards: EXC 2044: Mathematik Münster: Dynamik – Geometrie – Struktur, Advances in Mean Curvature Flow: Theory and Applications, G8029469473