Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Nirenberg problem of prescribed Gauss curvature on $S^{2}$   We introduce a new perspective on the classical Nirenberg problem of
understanding the possible Gauss curvatures of metrics on $S^{2}$ conformal to
the round metric. A key tool is to employ the smooth Cheeger-Gromov compactness
theorem to obtain general and essentially sharp a priori estimates for Gauss
curvatures $K$ contained in naturally defined stable regions. We prove that in
such stable regions, the map $u \rightarrow K_{g}$, $g = e^{2u}g_{+1}$ is a
proper Fredholm map with well-defined degree on each component. This leads to a
number of new existence and non-existence results. We also present a new proof
and generalization of the Moser theorem on Gauss curvatures of even conformal
metrics on $S^{2}$.
  In contrast to previous work, the work here does not use any of the
Sobolev-type inequalities of Trudinger-Moser-Aubin-Onofri.

13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
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The categories of this paper are: eess
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
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The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
25
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: solv-int