Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Compactness theorems of gradient Ricci solitons   In this paper, we prove the compactness theorem for gradient Ricci solitons.
Let $(M_{\alpha}, g_{\alpha})$ be a sequence of compact gradient Ricci solitons
of dimension $n\geq 4$, whose curvatures have uniformly bounded
$L^{\frac{n}{2}}$ norms, whose Ricci curvatures are uniformly bounded from
below with uniformly lower bounded volume and with uniformly upper bounded
diameter, then there must exists a subsequence $(M_{\alpha}, g_{\alpha})$
converging to a compact orbifold $(M_{\infty}, g_{\infty})$ with finitly many
isolated singularities, where $g_{\infty}$ is a gradient Ricci soliton metric
in an orbifold sense.

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The categories of this paper are: physics
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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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: funct-an
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The categories of this paper are: cond-mat
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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: bayes-an
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: hep-ph