A general regularity theory for weak mean curvature flow

Abstract

We give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard's regularity theorem in the sense that the special time-independent case reduces to Allard's theorem.

60 pages

References in corpus (4)

• Evolution of curves and surfaces by mean curvature
• A second derivative Hölder estimate for weak mean curvature flow
• Existence and regularity of mean curvature flow with transport term in higher dimensions
• Minimal surfaces and mean curvature flow

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• On the existence of canonical multi-phase Brakke flows
• Consequences of strong stability of minimal submanifolds
• A short proof of Allard's and Brakke's regularity theorems
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• Gradient estimates for mean curvature flow with Neumann boundary conditions
• An existence theorem for Brakke flow with fixed boundary conditions
• An example of a mean-convex mean curvature flow developing infinitely many singular epochs
• End-time regularity theorem for Brakke flows
• Brakke Regularity for the Allen-Cahn Flow
• Hausdorff Stability of the Round Two-Sphere Under Small Perturbations of the Entropy
• Lectures on mean curvature flow of surfaces
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• Existence of BV flow via elliptic regularization
• Allen-Cahn Approximation of Mean Curvature Flow in Riemannian manifolds II, Brakke's flows
• Quantitative Stratification and the Regularity of Mean Curvature Flow
• Epsilon-regularity for the Brakke flow with boundary