Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bubble tree of a class of conformal mappings and applications to
  Willmore functional   We develop a bubble tree construction and prove compactness results for
$W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with
varying conformal structures whose limit may degenerate, in a compact
Riemannian manifold with uniformly bounded areas and Willmore energies. The
compactness property is applied to construct Willmore type surfaces in compact
Riemannian manifolds. This includes (a) existence of a Willmore 2-sphere in
${\mathbb S}^n$ with at least 2 nonremovable singular points (b) existence of
minimizers of the Willmore functional with prescribed area in a compact
manifold $N$ provided (i) the area is small when genus is 0 and (ii) the area
is close to that of the area minimizing surface of Schoen-Yau and
Sacks-Uhlenbeck in the homotopy class of an incompressible map from a surface
of positive genus to $N$ and $\pi_2(N)$ is trivial (c) existence of smooth
minimizers of the Willmore functional if a Douglas type condition is satisfied.

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The categories of this paper are: cond-mat
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The categories of this paper are: eess
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The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
10
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: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat