Calculus of Variations and Geometric Measure Theory

R. Buzano

Monotone Volume Formulas for Geometric Flows

created by muller on 12 Jun 2018
modified on 13 Jun 2022


Published Paper

Inserted: 12 jun 2018
Last Updated: 13 jun 2022

Journal: J. Reine Angew. Math. (Crelle's Journal)
Volume: 643
Year: 2010
Doi: 10.1515/crelle.2010.044

ArXiv: 0905.2328 PDF

Published under previous name Reto Müller (please cite as Müller). Name changed here in order to import to author page correctly.


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.