Complete constant mean curvature hypersurfaces in Euclidean space of dimension four or higher
Complete constant mean curvature hypersurfaces in Euclidean space of dimension four or higher
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四维或更高维欧几里得空间中的完全常平均曲率超曲面
DOI：
10.1353/ajm.2021.0030
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发表时间：
2017
影响因子：
1.7
通讯作者：
N. Kapouleas
中科院分区：
文献类型：
--
作者：
Christine Breiner;N. Kapouleas
摘要：本文给出了浸入欧氏$（n+1）$-空间中的完备、光滑、常平均曲率的有限拓扑型超曲面（简称CMC $n$-超曲面）的一般构造.更确切地说，我们的构造将欧氏$（n+1）$-空间中的某些图转换为具有渐近Delaunay端点的CMC $n$-超曲面，这分为两步：首先，对给定图进行适当的小扰动，将其顶点替换为圆形球面区域，将其边和射线替换为Delaunay片，从而构造出一族初始光滑超曲面。然后扰动其中一个初始超曲面以产生所需的CMC $n$-超曲面，该超曲面取决于图的给定扰动族和一个小的绝对值参数$\underline{\tau}$。这种构造是非常普遍的，因为满足所需条件的图的丰富性，因为它不依赖于对称性要求。对于任何给定的k\ge2 $和n\ge3 $，它允许我们实现无限多个拓扑类型作为CMC $n$-超曲面在$\Bbb{R}^{n+1}$中有$k$结束。此外，对于每一种情况，都有大量的例子反映了可用图表的丰富性。这与已知的例子形成鲜明对比，这些例子据我们所知都是通过ODE方法获得的（广义）圆柱形，并且是紧凑的或两端的。此外，我们构造嵌入的例子时，$k\ge3$，其中可能的拓扑类型的数量为每个$k$是有限的，但往往$\infty$作为$k\到\infty$。最后，我们注意到，在正在进行的工作中，我们将这些结果扩展到构建无限多个拓扑类型的封闭（浸入）的例子，每个$n\ge3$。此外，对于每个$n\ge3$和$k\ge6$，我们构造了无穷多个具有$k$端的嵌入完备示例的拓扑类型。
abstract:In this article we provide a general construction when $n\ge3$ for immersed in Euclidean $(n+1)$-space, complete, smooth, constant mean curvature hypersurfaces of finite topological type (in short CMC $n$-hypersurfaces). More precisely our construction converts certain graphs in Euclidean $(n+1)$-space to CMC $n$-hypersurfaces with asymptotically Delaunay ends in two steps: First appropriate small perturbations of the given graph have their vertices replaced by round spherical regions and their edges and rays by Delaunay pieces so that a family of initial smooth hypersurfaces is constructed. One of the initial hypersurfaces is then perturbed to produce the desired CMC $n$-hypersurface which depends on the given family of perturbations of the graph and a small in absolute value parameter $\underline{\tau}$. This construction is very general because of the abundance of graphs which satisfy the required conditions and because it does not rely on symmetry requirements. For any given $k\ge2$ and $n\ge3$ it allows us to realize infinitely many topological types as CMC $n$-hypersurfaces in $\Bbb{R}^{n+1}$ with $k$ ends. Moreover for each case there is a plethora of examples reflecting the abundance of the available graphs. This is in sharp contrast with the known examples which in the best of our knowledge are all (generalized) cylindrical obtained by ODE methods and are compact or with two ends. Furthermore we construct embedded examples when $k\ge3$ where the number of possible topological types for each $k$ is finite but tends to $\infty$ as $k\to\infty$. Finally we remark that in ongoing work, we extend these results to construct infinitely many topological types of closed (immersed) examples for each $n\ge3$. Moreover, for each $n\ge3$ and $k\ge6$, we construct infinitely many topological types of embedded complete examples with $k$ ends.
影响因子：
2.5
作者：
W. Hsiang
通讯作者：
W. Hsiang