Complete Constant Mean Curvature Hypersurfaces in Euclidean space of dimension four or higher

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 $\mathbb R^{n+1}$ 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 $k\to\infty$.