Singular set and curvature blow-up rate of the level set flow

TL;DR

This paper characterizes the nature of singularities in the level set flow under 2-convexity, linking curvature blow-up rates to geometric shrinkage patterns and regularity of the arrival time function.

Contribution

It establishes a precise equivalence between type I singularities and specific geometric shrinking behaviors in the level set flow under 2-convexity.

Findings

Type I singularities occur only when the flow shrinks to a round point or a C^1 curve.

The arrival time function is C^2 near critical points if and only if it satisfies a Lojasiewicz inequality.

The results connect curvature blow-up rates with geometric and analytical regularity conditions.

Abstract

Under certain conditions such as the $2$-convexity, a singularity of the level set flow is of type I (in the sense that the rate of curvature blow-up is constrained before and after the singular time) if and only if the flow shrinks to either a round point or a $C^{1}$ curve near that singular point. Analytically speaking, the arrival time is $C^{2}$ near a critical point if and only if it satisfies a Lojasiewicz inequality near the point.