Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hyperbolic manifolds with convex boundary   Let $(M, \partial M)$ be a compact 3-manifold with boundary, which admits a
convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$
such that the boundary is smooth and strictly convex. We show that the induced
metrics on the boundary are exactly the metrics with curvature $K>-1$, and that
the third fundamental forms of $\dr M$ are exactly the metrics with curvature
$K<1$, for which contractible closed geodesics have length $L>2\pi$. Each is
obtained exactly once.
  Other related results describe existence and uniqueness properties for other
boundary conditions, when the metric which is achieved on $\dr M$ is a linear
combination of the first, second and third fundamental forms.

6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
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The categories of this paper are: stat
29
The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: physics
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs