The Limit of The Inverse Mean Curvature Flow on a Torus

Brian Harvie
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Abstract

For an $H>0$ rotationally symmetric embedded torus $N_{0} \subset \mathbb{R}^{3}$, evolved by Inverse Mean Curvature Flow, we show that the total curvature $|A|$ remains bounded up to the singular time $T_{\max}$. We then show convergence of the $N_{t}$ to a $C^{1}$ rotationally symmetric embedded torus $N_{T_{\max}}$ as $t \rightarrow T_{\max}$ without rescaling. Later, we observe a scale-invariant $L^{2}$ energy estimate on any embedded solution of the flow in $\mathbb{R}^{3}$ that may be useful in ruling out curvature blowup near singularities in general.
环面上逆平均曲率流的极限
对于一个$H>0$旋转对称嵌入环面$N_{0} \subset \mathbb{R}^{3}$,由逆平均曲率流进化,我们证明了总曲率$|A|$仍然有界到奇异时间$T_{\max}$。然后,我们将$N_{t}$收敛到$C^{1}$旋转对称嵌入环面$N_{T_{\max}}$为$t \rightarrow T_{\max}$,而无需重新缩放。随后,我们观察到$\mathbb{R}^{3}$中流动的任何嵌入解的尺度不变$L^{2}$能量估计,这可能有助于在一般情况下排除奇点附近的曲率爆炸。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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