Monotone volume formulas for geometric flows

We consider a closed manifold M with a Riemannian metric g(ij)(t) evolving by partial derivative(t)g(ij) = -2S(ij) where S(ij)(t) is a symmetric two-tensor on (M, g(t)). We prove that if S(ij) satisfies the tensor inequality D(S(ij); X) >= 0 for all vector fields X on M, where D(S(ij), 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 volume partial derivative(t)g(ij) = -2S(ij). In the case where S(ij) = R(ij), the Ricci curvature of M, the result corresponds to Perelman's well-known reduced volume monotonicity for the Ricci flow presented in [12]. 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 [8], the Ricci flow coupled with harmonic map heat flow presented in [11], and the mean curvature flow in Lorentzian manifolds with nonnegative sectional curvatures. With our approach, we find new monotonicity formulas for these flows.