§ 5.14 Rotational degenerate neckpinch solutions1E4AD3904D39

(Angenent–Velázquez type-II singularities) Let $n\geq 2$, let $m\geq 3$ be odd, and let $K>0$. Choose a compact rotational target hypersurface in $\mathbb R^{n+1}$, smooth away from a cusp at the origin, whose radius satisfies $u_*(x)\sim \sqrt K\, x^{m/2}$ as $x\downarrow 0$ and whose other tip closes smoothly. Angenent and Velázquez construct smooth compact rotational mean curvature flows

with smooth caps and positive radius between them. Each regular slice is an embedded sphere, hence complete and proper. The constructed flows remain close to the chosen target outside a small neighborhood of the origin as the construction interval is shortened. They are not supplied as ancient solutions or as solitons, and no closed formula for $u(x,t)$ is given.

Set $\xi _*=(2(n-1)/K)^{1/m}$. At the left tip,

Here the curvature coefficient is for the trace mean curvature, equivalently the magnitude $|\vec H|$ at this convex tip. Rotational smoothness makes the $n$ principal curvatures equal at the tip, so $|A|_{\rm tip}=|\vec H|_{\rm tip}/\sqrt n$. The maximum of $|A|$ is attained there; its blow-up rate is type II. The parabolic rescalings at the origin converge smoothly on compact sets to $S^{n-1}_{\sqrt{2(n-1)}}\times \mathbb R$, whereas rescaling about the tip by $A_{\max }(t)=\max _{M_ t}|A|$ yields the rotational translating bowl with tip $|A|=1$. This last normalization has translation speed $\sqrt n$; rescaling instead by $|\vec H|_{\rm tip}$ gives the speed-one bowl.

Under the intermediate scaling $x\mapsto x/(T-t)^{1/m}$ and $r\mapsto r/(T-t)^{1/2}$, the limiting profile is

This last expression is an asymptotic limiting profile, not an exact formula for the evolving compact hypersurface. Source: [AV97], Introduction, items 1–5, printed pp. 16–17, supplied PDF pp. 2–3; the construction is justified in Sections 3–7. The even-$m$ modifications and the formal $m=2$ discussion are not included in this entry. The illustration selects $n=2$, $m=3$, $K=1$, and writes $s=T-t$. Its time sequence uses one spatial scale to show the left tip approaching the origin while the outer cap persists; the final cusp target is not a smooth time slice. For the profile near the origin, item 5 of the Introduction gives

on bounded $y$ intervals. This leading Hermite correction explains the small bulge and neck in the cylindrical region. The rotatable compact slices blend the asymptotic profiles with a numerically integrated bowl cap and a chosen outer cap: they are illustrative schematics, not a computed solution of the compact MCF initial-value problem, and their displayed times carry no certified finite-time accuracy. Separate views compare the meridians and the three blow-up limits; the intermediate view uses unequal axial and radial scaling. The construction does not classify all type-II singularities or impose global convexity on every allowed target.