§ 2.9 Round shrinking spheres (Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017)2643D15A6912

For an integer $n\geq 1$, an extinction time $T\in \mathbb R$, and a center $a\in \mathbb R^{n+1}$, the exact family is

The centered normalized slice is $S^ n(\sqrt{2n})$ at $t=-1$.

• These are smooth compact embedded hypersurfaces without boundary, complete and proper, with topology $S^ n$ and full rotational symmetry. The case $n=1$ is the shrinking circle in the plane.

• With outward normal and $H=\operatorname {div}\nu $, all principal curvatures are $1/\sqrt{2n(T-t)}$. Thus the flow is strictly convex, $H=\sqrt{n/(2(T-t))}$ and $|A|^2=1/(2(T-t))$. It becomes extinct at $a$ in a Type I singularity at time $T$.

• The round sphere is entropy stable and $F$-stable, where $F$-stability allows the Gaussian center and scale to vary. These are the stability notions in [CM12], Theorems 0.12 and 0.16; stability of Gaussian area with center and scale fixed is a different question.

• The normalized Gaussian integral gives

  \[ \lambda (S^ n(\sqrt{2n})) =(4\pi )^{-n/2}(2n)^{n/2}e^{-n/2}|S^ n(1)|. \]

  In particular, a two-sphere has entropy $4/e$.

The three-dimensional picture is the exact surface for $n=2$; for other dimensions a round profile represents the symmetry reduction.