Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence of immersed spheres minimizing curvature functionals in
  non-compact 3-manifolds   We study curvature functionals for immersed 2-spheres in non-compact,
three-dimensional Riemannian manifold $(M,h)$ without boundary. First, under
the assumption that $(M,h)$ is the euclidean 3-space endowed with a
semi-perturbed metric with perturbation small in $C^1$ norm and of compact
support, we prove that if there is some point $\bar{x} \in M$ with scalar
curvature $R^M(\bar{x})>0$ then there exists a smooth embedding $f:S^2
\hookrightarrow M$ minimizing the Willmore functional $1/4\int |H|^2$, where
$H$ is the mean curvature. Second, assuming that $(M,h)$ is of bounded geometry
(i.e. bounded sectional curvature and strictly positive injectivity radius) and
asymptotically euclidean or hyperbolic we prove that if there is some point
$\bar{x} \in M$ with scalar curvature $R^M(\bar{x})>6$ then there exists a
smooth immersion $f:S^2 \hookrightarrow M$ minimizing the functional $\int
(1/2|A|^2+1)$, where $A$ is the second fundamental form. Finally, adding the
bound $K^M \leq 2$ to the last assumptions, we obtain a smooth minimizer $f:S^2
\hookrightarrow M$ for the functional $\int (1/4|H|^2+1)$. The assumptions of
the last two theorems are satisfied in a large class of 3-manifolds arising as
spacelike timeslices solutions of the Einstein vacuum equation in case of null
or negative cosmological constant.

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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: bayes-an
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The categories of this paper are: q-alg
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The categories of this paper are: econ
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: physics