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arXiv · 2110.06027

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

Reto Buzano·Huy The Nguyen·Mario B. Schulz
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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.

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Reto Buzano, Huy The Nguyen, Mario B. Schulz. 2021-10-12. Noncompact self-shrinkers for mean curvature flow with arbitrary genus. https://doi.org/10.1515/crelle-2024-0073

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