Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complete Constant Mean Curvature Hypersurfaces in Euclidean space of
  dimension four or higher   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 $k\to\infty$.

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The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph