Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Conformal surface embeddings and extremal length   Given two Riemann surfaces with boundary and a homotopy class of topological
embeddings between them, there is a conformal embedding in the homotopy class
if and only if the extremal length of every simple multi-curve is decreased
under the embedding. Furthermore, the homotopy class has a conformal embedding
that misses an open disk if and only if extremal lengths are decreased by a
definite ratio. This ratio remains bounded away from one under covers.

16
The categories of this paper are: stat
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg