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Under certain conditions such as the $2$-convexity, a singularity of thelevel set flow is of type I (in the sense that the rate of curvature blow-up isconstrained before and after the singular time) if and only if the flow shrinksto 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 andonly if it satisfies a Lojasiewicz inequality near the point.

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