Sharp dynamical isoperimetric principle for mean curvature flow

In this paper, we establish a sharp dynamical isoperimetric principle for weak mean curvature flow. We prove that, among weak mean curvature flows with fixed area of initial smooth closed hypersurfaces, the standard smoothly shrinking spherical mean curvature flow uniquely maximizes the extinction time. More precisely, for the level set flow $K_t$ starting from the boundary of a smooth bounded domain $\Omega \subset {\mathbb{R}}^{n+1}$, as well as for the integral Brakke flow $\{{\mu }_t\}$ with initial Radon measure ${\mu }_0={\mathcal{H}}^n└\partial \Omega$, we establish the corresponding optimal extinction time estimates \beginequation* T_{\rm ext}(Ω),\, T^B_{\rm ext}(Ω) \leq \frac12n \left(\fracP( Ω){|{\mathbbS^n}|}\right)^{\frac2n}, \endequation* where $P(\Omega )$ is the perimeter of $\Omega$ representing the area of $\partial \Omega$, and the equality holds if and only if $\Omega$ is a round ball and the flow is the standard multiplicity-one smoothly self-shrinking round sphere. In particular, we obtain the sharp $L^p$-estimates for the arrival time function of a smooth bounded mean convex domain. In addition, we also establish the sharp isoperimetric inequality for the parabolic measure of the space-time track filling $X$ of outward minimizing level set flow starting from the boundary of a smooth bounded domain $\Omega \subset {\mathbb{R}}^{n+1}$: \beginequation* \mathcal H_par^n+2(X) \leq \fracπ {2n(n+2)^2|{\mathbbS^n}|^{{\frac2n}}} P(Ω)^{\fracn+2n}, \endequation* where the equality holds if and only if $\Omega$ is a round ball and the flow is standard smoothly shrinking round sphere.