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A strong Frankel theorem for shrinkers

Part of: Classical differential geometry Global differential geometry

Published online by Cambridge University Press:  19 August 2025

Tobias Holck Colding  and
William P. Minicozzi II
Tobias Holck Colding
Affiliation:
Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA colding@math.mit.edu
William P. Minicozzi II
Affiliation:
Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA minicozz@math.mit.edu
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Abstract

We prove a strong Frankel theorem for mean curvature flow shrinkers in all dimensions: Any two shrinkers in a sufficiently large ball must intersect. In particular, the shrinker itself must be connected in all large balls. The key to the proof is a strong Bernstein theorem for incomplete stable Gaussian surfaces.

Keywords

mean curvature flowself-shrinkersGaussian area

MSC classification

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 53A07: Higher-dimensional and -codimensional surfaces in Euclidean $n$-space 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 5 , May 2025 , pp. 947 - 958
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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