论文标题
高编纂平均曲率流量的凸度估计值
Convexity Estimates for High Codimension Mean Curvature Flow
论文作者
论文摘要
我们考虑了$ \ Mathbb {r}^{n+k} $的平滑$ n $ dimensional submanifolds的平均曲率的演变，这些曲率是紧凑且四边形捏合的。我们将主要对高复合的流动感兴趣，即$ k \ geq 2 $。我们证明我们的子序是渐近的凸，这是主要平均曲率方向在主要平均曲率方向上的第一个特征值，比平均曲率向量的速度严格速度较慢。我们使用此凸度估计值表明，在流动的奇异时间，存在重新分组，该重新收敛到平滑的编成一个限制流，该流量是凸出并通过翻译移动的。
We consider the evolution by mean curvature of smooth $n$-dimensional submanifolds in $\mathbb{R}^{n+k}$ which are compact and quadratically pinched. We will be primarily interested in flows of high codimension, the case $k\geq 2$. We prove that our submanifold is asymptotically convex, that is the first eigenvalue of the second fundamental form in the principal mean curvature direction blows up at a strictly slower rate than the mean curvature vector. We use this convexity estimate to show that at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow which is convex and moves by translation.
