Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complete Embedded Harmonic Surfaces In $\mathbb{R}^3$   Embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$ are
reasonably well understood: From far away, they look like intersecting
catenoids and planes, suitably desingularized. We consider the larger class of
harmonic embeddings in $\mathbb{R}^{3}$ of compact Riemann surfaces with
finitely many punctures where the harmonic coordinate functions are given as
real parts of meromorphic functions. This paper is motivated by two outstanding
features of such surfaces: They can have highly complicated ends, and they
still have total Gauss curvature being a multiple of $2\pi$. This poses the
double challenge to construct and classify examples of fixed total Gauss
curvature. Our results include
  - a classification of embedded harmonic ends of small total curvature,
  - the construction of examples of embedded ends of arbitrarily large total
curvature,
  - a classification of complete embedded harmonic surfaces of small total
curvature in the spirit of the corresponding classification of minimal surfaces
of small total curvature, and
  - the largely experimental construction of complete embedded harmonic
surfaces with non-trivial topology that incorporate some of the new harmonic
ends.

19
The categories of this paper are: eess
29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn