# Brakke's theorem for gap in entropy between self-shrinkers

In their paper The round sphere minimizes entropy among closed self shrinkers, Colding-Ilmanen-Minicozzi-White state "It follows from Brakke's theorem that $\mathbb{R}^n$ has the least entropy of any self-shrinker and, in fact, there is a gap to the next lowest." Why is this?

Here the entropy of a hypersurface $\Sigma^n\subset\mathbb{R}^{n+1}$ is defined as $$\lambda(\Sigma)=\sup_{\substack{t_0>0\\ x_0\in\mathbb{R}^{n+1}}}\frac{1}{(4\pi t_0)^{n/2}}\int_\Sigma e^{-|x-x_0|^2/4t_0}\,d\mathcal{H}^n.$$ In their paper Generic mean curvature flow I, Colding-Minicozzi show that for a (smooth) self-shrinker with polynomial volume growth which does not isometrically split off a line, the supremum is achieved at $t_0=1$ and $x_0=0.$ Here a self-shrinker is a surface $\Sigma_{-1}$ such that $\Sigma_t=\sqrt{-t}\cdot\Sigma_{-1}$ is a mean curvature flow. (It is obvious that the entropy is constant along a self-shrinking flow. For any mean curvature flow it is non-increasing, as an easy consequence of Huisken's monotonicity.)

Brakke's book is available here. His main regularity theorem is on page 210, and his result on existence of the flow for a general initial varifold is in chapter 4. I can't see why either is helpful here.

• Does that follows from White's regularity theorem that there is $\epsilon >0$ such that if $\lambda(\Sigma)< 1+ \epsilon$ then $\Sigma$ is smooth at $0$? – Arctic Char Feb 16 '14 at 10:35
• This does follow from White's theorem; under the entropy assumption we get a uniform bound on the second fundamental form of the shrinker (i.e. uniform in spacetime), but as the shrinker moves by $\Sigma_t=\sqrt{-t}\cdot\Sigma_{-1}$ on $t<0$, the second fundamental form is blowing up as $t\nearrow 0$ unless it vanishes everywhere, a contradiction unless the second fundamental form does vanish everywhere, in which case it is a plane. But I have seen this fact regularly attributed to Brakke's theorem and not to White's, so I think something else is going on here. – protocokodo Mar 2 '14 at 20:29