Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bifurcation for Minimal Surface Equation in Hyperbolic $3$-Manifolds   Initiated by the work of Uhlenbeck in late 1970s, we study questions about
the existence, multiplicity and asymptotic behavior for minimal immersions of
closed surface in some hyperbolic three-manifold, with prescribed conformal
structure on the surface and second fundamental form of the immersion. We prove
several results in these directions. In particular, we determine when exactly
the solution is unique and when multiple solutions appear. Moreover, we analyze
in detail the asymptotic behavior of the solutions when (and how) blowing up
might occur. Interestingly the blow-up analysis exhibit different behaviors
when the surface is of genus two or greater. Furthermore, we extend this
program to consider similar problems where the total extrinsic curvature is
prescribed and we prove an existence result.

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The categories of this paper are: nucl-th
16
The categories of this paper are: stat
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
26
The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ