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.File | Dimensione | Formato | |
---|---|---|---|
Noncompact selfshrinkers 10.1515_crelle-2024-0073.pdf
Solo gestori archivio
Descrizione: Publisher's
Tipologia:
Versione editoriale (Publisher’s layout)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
12.75 MB
Formato
Adobe PDF
|
12.75 MB | Adobe PDF | Visualizza/Apri |
10.1515_crelle-2024-0073_compressed.pdf
Solo gestori archivio
Tipologia:
Versione editoriale (Publisher’s layout)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
321.02 kB
Formato
Adobe PDF
|
321.02 kB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione