论文标题
通过手术通过平均曲率流动的球形二次捏式超曲面
Quadratically pinched hypersurfaces of the sphere via mean curvature flow with surgery
论文作者
论文摘要
我们在$ \ mathbb s_k^{n + 1} $中研究平均曲率流，在分段曲率$ k> 0 $的圆形范围内，在二次弯曲条件下$ | | a | a | a | a |^{2} <\ frac {1} \ frac {3} {5} h^{2}+\ frac {8} {3} k $时$ n = 3 $。这种情况与著名的西蒙斯定理有关，该定理指出，满足$ \ vert a \ vert^2 <nk $的唯一最小的超曲面是完全地球的超球。它与两个凸度有关，但与两个凸度不同。值得注意的是，与两个凸度相比，它允许平均曲率更改符号。我们表明，捏合条件通过平均曲率流得到保留，并获得曲率的圆柱估计值和相应的点衍生估计值。结果，我们发现在高曲率区域中，流量变为均匀凸的或定量圆柱形。这使我们能够应用Huisken和Sinestrari开发的手术设备。我们得出的结论是，任何顺利，适当地融入了$ \ mathcal $ \ Mathcal {m} $ $ \ mathbb s_k^{n+1} $满足捏合条件的$ \ \ \ \ \玛理布bb s^n $或连接的$ \ n $ $ \ mathb s^n $连接数量s^{n-1} $。如果嵌入了$ \ Mathcal M $，则它将界限1个手机。当$ n \ ge 4 $时，结果很清晰。
We study mean curvature flow in $\mathbb S_K^{n+1}$, the round sphere of sectional curvature $K>0$, under the quadratic curvature pinching condition $|A|^{2} < \frac{1}{n-2} H^{2} + 4 K$ when $n\ge 4$ and $|A|^{2} < \frac{3}{5}H^{2}+\frac{8}{3}K$ when $n=3$. This condition is related to a famous theorem of Simons, which states that the only minimal hypersurfaces satisfying $\vert A\vert^2<nK$ are the totally geodesic hyperspheres. It is related to but distinct from two-convexity. Notably, in contrast to two-convexity, it allows the mean curvature to change sign. We show that the pinching condition is preserved by mean curvature flow, and obtain a cylindrical estimate and corresponding pointwise derivative estimates for the curvature. As a result, we find that the flow becomes either uniformly convex or quantitatively cylindrical in regions of high curvature. This allows us to apply the surgery apparatus developed by Huisken and Sinestrari. We conclude that any smoothly, properly, isometrically immersed hypersurface $\mathcal{M}$ of $\mathbb S_K^{n+1}$ satisfying the pinching condition is diffeomorphic to $\mathbb S^n$ or the connected sum of a finite number of copies of $\mathbb S^1\times \mathbb S^{n-1}$. If $\mathcal M$ is embedded, then it bounds a 1-handlebody. The results are sharp when $n\ge 4$.
