Regularity of elliptic and parabolic systems
Tobias Holck Colding, William P. Minicozzi II
TL;DR
This paper proves the uniqueness of cylindrical blowups in mean curvature flow across all dimensions and codimensions, leading to regularity results for the singular set in complex systems.
Contribution
It establishes the first proof of uniqueness of cylindrical blowups for mean curvature flow in all dimensions and codimensions, addressing a major open problem.
Findings
Uniqueness of cylindrical blowups in mean curvature flow
Regularity of the singular set for the system
Applicability to all dimensions and codimensions
Abstract
We show uniqueness of cylindrical blowups for mean curvature flow in all dimension and all codimension. Cylindrical singularities are known to be the most important; they are the most prevalent in any codimension. Mean curvature flow in higher codimension is a nonlinear parabolic system where many of the methods used for hypersurfaces do not apply and uniqueness of cylindrical blowups remained a major open problem. Our results imply regularity of the singular set for the system.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations · Advanced Mathematical Modeling in Engineering