Monotone volume formulas for geometric flows

Abstract

We consider a closed manifold M with a Riemannian metric gij(t) evolving by ∂tgij = –2Sij where Sij(t) is a symmetric two-tensor on (M, g(t)). We prove that if Sij satisfies the tensor inequality 𝒟(Sij, X) ≧ 0 for all vector fields X on M, where 𝒟(Sij, X) is defined in (1.6), 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 ∂tgij = –2Sij. In the case where Sij = Rij, the Ricci curvature of M, the result corresponds to Perelman's well-known reduced volume monotonicity for the Ricci flow presented in [The entropy formula for the Ricci flow and its geometric applications, 2002]. 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 developed in [Evolution of an extended Ricci flow system, AEI Potsdam, 2005], the Ricci flow coupled with harmonic map heat flow presented in [Müller, The Ricci flow coupled with harmonic map heat flow, ETH Zürich, 2009], and the mean curvature flow in Lorentzian manifolds with nonnegative sectional curvatures. With our approach, we find new monotonicity formulas for these flows.