[Paper Review] The extension problem of the mean curvature flow (I)

Haozhao Li, Bing Wang|arXiv (Cornell University)|Aug 9, 2016

Geometric Analysis and Curvature Flows44 references3 citations

TL;DR

This paper proves that the mean curvature blows up at the first finite singular time for any closed, smooth, embedded mean curvature flow in ℝ³, resolving a long-standing conjecture. Using rescaling and weak compactness techniques, the authors establish multiplicity-one convergence of the rescaled flow to a plane, contradicting higher multiplicity limits via parabolic Harnack and stability estimates.

ABSTRACT

We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R^3.

Motivation & Objective

• To resolve the conjecture that mean curvature must blow up at the first finite singular time in 3D mean curvature flow.
• To establish the extension problem for mean curvature flow without assuming initial convexity.
• To prove that the rescaled mean curvature flow converges to a plane with multiplicity one under bounded curvature.
• To develop a two-sided, long-time pseudolocality theorem for non-graphical flows.
• To show that higher multiplicity limits lead to contradictions via parabolic Harnack and stability arguments.

Proposed method

• Rescale the mean curvature flow to transform the singular time into infinity, converting curvature blow-up into decay to zero.
• Establish a two-sided, long-time pseudolocality theorem under bounded mean curvature and area doubling, removing the need for local graph conditions.
• Prove an energy concentration property using the pseudolocality theorem to control curvature concentration.
• Apply weak compactness to extract a limit flow that is both minimal and self-shrinking, hence a plane.
• Use uniform estimates on positive solutions to parabolic equations on compact subsets of the limit surface.
• Leverage L²-stability of the limit plane across singular sets to rule out multiplicity greater than one.

Experimental results

Research questions

• RQ1 Does the mean curvature necessarily blow up at the first finite singular time for a closed embedded mean curvature flow in ℝ³?
• RQ2 Can the extension problem for mean curvature flow be resolved without assuming convexity of the initial hypersurface?
• RQ3 Is the limit of the rescaled mean curvature flow necessarily a plane with multiplicity one?
• RQ4 Can a two-sided pseudolocality theorem be established for non-graphical, bounded curvature mean curvature flows?
• RQ5 Can the existence of a positive solution to a parabolic equation on a limit surface with multiplicity >1 be ruled out?

Key findings

• The mean curvature blows up at the first finite singular time T < ∞ for any closed, smooth, embedded mean curvature flow in ℝ³.
• The rescaled mean curvature flow converges weakly to a plane with multiplicity one, under the assumption that curvature remains bounded.
• A two-sided, long-time pseudolocality theorem is established for mean curvature flows with bounded H and area doubling, valid even without local graph conditions.
• Energy concentration is proven via the pseudolocality theorem, enabling weak compactness of the flow sequence.
• The existence of a positive solution to the parabolic equation on a limit surface with multiplicity >1 leads to a contradiction via parabolic Harnack and stability estimates.
• The limit surface is shown to be L²-stable, which rules out higher multiplicity limits, confirming multiplicity one convergence.