论文标题
圆环上平均曲率流的极限
The Limit of the Inverse Mean Curvature Flow on a Torus
论文作者
论文摘要
对于$ h> 0 $，旋转对称的嵌入式圆环$ n_ {0} \ subset \ subset \ mathbb {r}^{3} $，通过反平均值曲率流进行演变，我们显示总曲率$ | a | $保持到唯一的时间$ t _ {\ max} $。然后，我们将$ n_ {t} $的收敛性与$ c^{1} $旋转对称嵌入式圆环$ n_ {t _ {\ max}} $作为$ t \ rightArrow t _ {\ max} $而无需重新分组。稍后，我们观察到一个比例不变的$ l^{2} $能量估计在$ \ mathbb {r}^{3} $中的任何嵌入式解决方案上，这些解决方案可能有助于排除一般而言近乎奇异的曲率爆炸。
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.
