In this paper, we generalize Colding-Minicozzi's recent results about codimension-1 self-shrinkers for the mean curvature flow to higher codimension. In particular, we prove that the sphere is the only complete embedded connected -stable self-shrinker in with , polynomial volume growth, flat normal bundle and bounded geometry. We also discuss some properties of symplectic self-shrinkers, proving that any complete symplectic self-shrinker in with polynomial volume growth and bounded second fundamental form is a plane. As a corollary, we show that there is no finite time Type I singularity for symplectic mean curvature flow, which has been proved by Chen-Li using different method. We also study Lagrangian self-shrinkers and prove that for Lagrangian mean curvature flow, the blow-up limit of the singularity may be not F-stable.

Self-shrinkers for the mean curvature flow in arbitrary codimension / Arezzo, Claudio; J., Sun. - In: MATHEMATISCHE ZEITSCHRIFT. - ISSN 0025-5874. - 274:3-4(2013), pp. 993-1027. [10.1007/s00209-012-1104-y]

Self-shrinkers for the mean curvature flow in arbitrary codimension

AREZZO, Claudio;
2013-01-01

Abstract

In this paper, we generalize Colding-Minicozzi's recent results about codimension-1 self-shrinkers for the mean curvature flow to higher codimension. In particular, we prove that the sphere is the only complete embedded connected -stable self-shrinker in with , polynomial volume growth, flat normal bundle and bounded geometry. We also discuss some properties of symplectic self-shrinkers, proving that any complete symplectic self-shrinker in with polynomial volume growth and bounded second fundamental form is a plane. As a corollary, we show that there is no finite time Type I singularity for symplectic mean curvature flow, which has been proved by Chen-Li using different method. We also study Lagrangian self-shrinkers and prove that for Lagrangian mean curvature flow, the blow-up limit of the singularity may be not F-stable.
2013
Self-shrinkers for the mean curvature flow in arbitrary codimension / Arezzo, Claudio; J., Sun. - In: MATHEMATISCHE ZEITSCHRIFT. - ISSN 0025-5874. - 274:3-4(2013), pp. 993-1027. [10.1007/s00209-012-1104-y]
File in questo prodotto:
File Dimensione Formato  
arezzosunmathz.pdf

non disponibili

Tipologia: Documento in Post-print
Licenza: NON PUBBLICO - Accesso privato/ristretto
Dimensione 720.76 kB
Formato Adobe PDF
720.76 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11381/2673259
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 23
  • ???jsp.display-item.citation.isi??? 19
social impact