论文标题
定量亚历山德罗夫定理和体积保留平均曲率流的渐近行为
Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow
论文作者
论文摘要
我们证明了Alexandrov定理的新定量版本，该版本指出，如果R^{n+1}中常规集的平均曲率接近l^{n} - sense中的常数，则该集合接近与Hausdorff距离的不相交球的结合。该结果比Alexandrov定理的先前量化更一般，并且使用它，我们能够证明，在R^2和R^3中，从一组有限的周边收敛到较弱的平均曲率流的较弱的解决方案，从一组有限的周围收敛到廉价球的不相交的均不相交。在这里，我们是指通过最小化运动方案获得的平坦流量。
We prove a new quantitative version of the Alexandrov theorem which states that if the mean curvature of a regular set in R^{n+1} is close to a constant in L^{n}-sense, then the set is close to a union of disjoint balls with respect to the Hausdorff distance. This result is more general than the previous quantifications of the Alexandrov theorem and using it we are able to show that in R^2 and R^3 a weak solution of the volume preserving mean curvature flow starting from a set of finite perimeter asymptotically convergences to a disjoint union of equisize balls, up to possible translations. Here by weak solution we mean a flat flow, obtained via the minimizing movements scheme.
