Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Monotone Volume Formulas for Geometric Flows   We consider a closed manifold M with a Riemannian metric g(t) evolving in
direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove
that if S satisfies a certain tensor inequality, then one can construct a
forwards and a backwards reduced volume quantity, the former being
non-increasing, the latter being non-decreasing along the flow. In the case
where S=Ric is the Ricci curvature of M, the result corresponds to Perelman's
well-known reduced volume monotonicity for the Ricci flow. Some other examples
are given in the second section of this article, the main examples and
motivation for this work being List's extended Ricci flow system, the Ricci
flow coupled with harmonic map heat flow and the mean curvature flow in
Lorentzian manifolds with nonnegative sectional curvatures. With our approach,
we find new monotonicity formulas for these flows.

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The categories of this paper are: stat
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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: cs
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The categories of this paper are: q-alg
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The categories of this paper are: ao-sci
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: physics
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The categories of this paper are: hep-th