AI 中文总结
本文为弱平均曲率流建立了尖锐动力等周原理，证明在初始面积固定的流中标准收缩球面唯一最大化消亡时间，并给出最优消亡时间估计及抛物测度等周不等式。
AI 中文摘要
本文针对弱平均曲率流建立了尖锐的动力等周原理。我们证明，在初始光滑闭超曲面面积固定的弱平均曲率流中，标准光滑收缩球面平均曲率流唯一地最大化消亡时间。更精确地说，对于从光滑有界域 $\Omega\subset\mathbb R^{n+1}$ 的边界出发的水平集流 $K_t$，以及具有初始 Radon 测度 $\mu_0=\mathcal H^n\llcorner\partial\Omega$ 的积分 Brakke 流 $\{\mu_t\}$，我们建立了相应的最优消亡时间估计 \begin{equation*} T_{\rm ext}(\Omega),\\, T^B_{\rm ext}(\Omega) \leq \frac{1}{2n} \left(\frac{P( \Omega)}{|{\mathbb{S}^n}|}\right)^{\frac{2}{n}}, \end{equation*} 其中 $P(\Omega)$ 是 $\Omega$ 的周长，表示 $\partial \Omega$ 的面积，等号成立当且仅当 $\Omega$ 是圆球且流是标准重数一光滑自相似收缩圆球。特别地，我们获得了光滑有界平均凸域的到达时间函数的尖锐 $L^p$ 估计。此外，我们还建立了从光滑有界域 $\Omega\subset \mathbb R^{n+1}$ 的边界出发的外向极小化水平集流的时空轨道填充 $X$ 的抛物测度的尖锐等周不等式： \begin{equation*} \mathcal H_{\mathrm{par}}^{n+2}(X) \leq \frac{\pi} {2n(n+2)^2|{\mathbb{S}^n}|^{{\frac{2}{n}}}} P(\Omega)^{\frac{n+2}{n}}, \end{equation*} 其中等号成立当且仅当 $\Omega$ 是圆球且流是标准光滑收缩圆球。
英文摘要
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 $Ω\subset\mathbb R^{n+1}$, as well as for the integral Brakke flow $\{μ_t\}$ with initial Radon measure $μ_0=\mathcal H^n\llcorner\partialΩ$, we establish the corresponding optimal extinction time estimates \begin{equation*} T_{\rm ext}(Ω),\, T^B_{\rm ext}(Ω) \leq \frac{1}{2n} \left(\frac{P( Ω)}{|{\mathbb{S}^n}|}\right)^{\frac{2}{n}}, \end{equation*} where $P(Ω)$ is the perimeter of $Ω$ representing the area of $\partial Ω$, and the equality holds if and only if $Ω$ 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 $Ω\subset \mathbb R^{n+1}$: \begin{equation*} \mathcal H_{\mathrm{par}}^{n+2}(X) \leq \fracπ {2n(n+2)^2|{\mathbb{S}^n}|^{{\frac{2}{n}}}} P(Ω)^{\frac{n+2}{n}}, \end{equation*} where the equality holds if and only if $Ω$ is a round ball and the flow is standard smoothly shrinking round sphere.
