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.

摘要：

在本文中，我们提供了一个通用构造，当 $n\ge3$ 沉浸在欧几里得 $(n+1)$-空间中时，有限拓扑类型的完整、平滑、恒定平均曲率超曲面（简称 CMC $n$-超曲面） . 更准确地说，我们的构造将欧几里得 $(n+1)$-空间中的某些图转换为 CMC $n$-超曲面，其中渐近 Delaunay 以两个步骤结束：首先，给定图的适当小扰动将其顶点替换为圆球区域，并且它们的边缘和射线由 Delaunay 块组成，从而构建了一系列初始平滑超曲面。然后扰动初始超曲面之一以产生所需的 CMC $n$-超曲面，这取决于图的给定扰动系列和绝对值较小的参数 $\underline{\tau}$。这种构造非常通用，因为满足所需条件的图非常丰富，而且它不依赖于对称性要求。对于任何给定的 $k\ge2$ 和 $n\ge3$，它允许我们在 $\Bbb{R}^{n+1}$ 中以 $k$ 结尾的 CMC $n$-超曲面形式实现无限多种拓扑类型。此外，对于每种情况，都有大量示例反映了可用图表的丰富性。这与已知示例形成鲜明对比，据我们所知，这些示例都是通过 ODE 方法获得的（广义）圆柱，并且是紧凑的或有两端的。此外，我们在 $k\ge3$ 时构建嵌入式示例，其中每个 $k$ 的可能拓扑类型的数量是有限的，但趋向于 $\infty$ 为 $k\to\infty$。最后我们指出，在正在进行的工作中，我们扩展这些结果，为每个 $n\ge3$ 构造无限多个拓扑类型的封闭（浸入式）示例。此外，对于每个 $n\ge3$ 和 $k\ge6$，我们构造了无限多个具有 $k$ 端的嵌入完整示例的拓扑类型。