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

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.