Calculus of Variations and Geometric Measure Theory
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R. Müller

Monotone Volume Formulas for Geometric Flows

created by muller on 24 Apr 2010
modified on 12 Jun 2018


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

ArXiv: 0905.2328 PDF


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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