Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gromov compactness theorem for stable curves   We give a proof of the Gromov compactness theorem using the language of
stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and
Manin) in general setting: An almost complex structure on a target manifold is
only continuous and can vary; the curves are only assumed to have fixed
``topological type'', in particular they can be non-closed and the complex
structures on them can vary arbitrarily.
  In connection with this, we study in \S 2 moduli spaces of nodal curves with
boundary and define a natural complex structure for such moduli spaces.
  We obtain an apriori estimate for pseudoholomorphic maps of ``long
cylinders'' (see \S 3), which gives a uniform description for degeneration of
complex structure on the curves and for the ``bubbling'' phenomenon. It also
implies the Hausdorff convergence of the curves.
  We also prove in \S 5 the compactness theorem for curves with boundary on
totally real submanifolds. For this ``boundary'' case we give appropriate
generalizations of all ``inner'' constructions and estimates.

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The categories of this paper are: nlin
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The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: chem-ph
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph
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The categories of this paper are: econ