Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Carath\'{e}odory balls and proper holomorphic maps on multiply-connected
  planar domains   In this paper, we will establish the inequivalence of closed balls and the
closure of open balls under the Carath\'{e}odory metric in some planar domains
of finite connectivity greater than $2$, and hence resolve a problem posed by
Jarnicki, Pflug and Vigu\'{e} in 1992. We also establish a corresponding result
for some pseudoconvex domains in $\mathbb{C}^n$ for $n \ge 2$.
  This result will follow from an explicit characterization (up to
biholomorphisms) of proper holomorphic maps from a non-degenerate
finitely-connected planar domain, $\Omega$, onto the standard unit disk
$\mathbb{D}$ which answers a question posed by Schmieder in 2005. Similar to
Bell and Kaleem's characterization of proper holomorphic maps in terms of
Grunsky maps (2008), our characterization of proper holomorphic maps from
$\Omega$ onto $\mathbb{D}$ is an analogous result to Fatou's famous result that
proper holomorphic maps of the unit disk onto itself are finite Blashcke
products.
  Our approach uses a harmonic measure condition of Wang and Yin (2017) on the
existence of a proper holomorphic map with prescribed zeros. We will see that
certain functions $\eta(\cdot,p)$ play an analogous role to the M\"{o}bius
transformations that fix $\mathbb{D}$ in finite Blaschke products. These
functions $\eta(\cdot,p)$ map $\Omega$ conformally onto the unit disk with
circular arcs (centered at 0) removed and map $p$ to $0$ and can be given in
terms of the Schottky-Klein prime function. We also extend a result of Grunsky
(1941) and hence introduce a parameter space for proper holomorphic maps from
$\Omega$ onto $\mathbb{D}$.

1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: adap-org
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The categories of this paper are: q-alg
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The categories of this paper are: plasm-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: q-bio
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The categories of this paper are: acc-phys
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: eess
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The categories of this paper are: cond-mat