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.

Noncompact self-shrinkers for mean curvature flow with arbitrary genus / Buzano, Reto; Nguyen, Huy The; Schulz, Mario B.. - In: JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK. - ISSN 0075-4102. - 88:(2025), pp. 35-52. [10.1515/crelle-2024-0073]

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

Schulz, Mario B.
2025-01-01

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.
2025
Buzano, Reto; Nguyen, Huy The; Schulz, Mario B.
Noncompact self-shrinkers for mean curvature flow with arbitrary genus / Buzano, Reto; Nguyen, Huy The; Schulz, Mario B.. - In: JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK. - ISSN 0075-4102. - 88:(2025), pp. 35-52. [10.1515/crelle-2024-0073]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/439558
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