Degenerate neckpinches at the first $N$ singular times

Abstract

For every integer $N\ge1$, we construct a smooth closed connected strictly mean-convex rotationally symmetric surface in $\mathbb{R}^3$ whose level set flow remains connected until extinction and has exactly one singular point at each of its first $N$ singular times. These singularities are degenerate neckpinches with the Angenent--Velázquez asymptotics: the leading deviation from the cylinder under parabolic rescaling is a cubic Hermite term.