## R. MÃ¼ller

# Monotone Volume Formulas for Geometric Flows

created by muller on 24 Apr 2010

modified on 12 Jun 2018

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BibTeX]

*Published Paper*

**Inserted:** 24 apr 2010

**Last Updated:** 12 jun 2018

**Journal:** J. Reine Angew. Math. (Crelle's Journal)

**Volume:** 2010

**Pages:** 39-57

**Year:** 2010

**Abstract:**

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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