论文标题
通过平均曲率流和凸面上的自由边界的平均曲率流进行收缩凸表面
Contracting convex surfaces by mean curvature flow with free boundary on convex barriers
论文作者
论文摘要
我们考虑欧几里德式凸面表面的平均曲率流量$ 3 $ - 空间，自由边界位于任意凸面屏障表面上，具有有界的几何形状。当初始表面充分凸出时，仅取决于屏障的几何形状，流动在有限的时间内将表面缩小到一个点。此外，该解决方案是渐近地缩小了半个球的缩小。这扩展了第二个维度，即脐带屏障到一般凸屏障的稳定性结果。我们介绍了一个新的扰动论点，以建立对流量的基本凸性和捏合估计。可以将我们的结果与霍斯肯的著名融合定理进行比较，以在Riemannian歧管中的凸出曲面的平均曲率流动。
We consider the mean curvature flow of compact convex surfaces in Euclidean $3$-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on the geometry of the barrier, the flow contracts the surface to a point in finite time. Moreover, the solution is asymptotic to a shrinking half-sphere lying in a half space. This extends, in dimension two, the convergence result of Stahl for umbilic barriers to general convex barriers. We introduce a new perturbation argument to establish fundamental convexity and pinching estimates for the flow. Our result can be compared to a celebrated convergence theorem of Huisken for mean curvature flow of convex hypersurfaces in Riemannian manifolds.
