We investigate a coupled system of the Ricci flow on a closed manifold M with the harmonic map flow of a map φ from M to some closed target manifold N , ∂ ∂t

We define several notions of singular set for Type I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of Naber [15]. As a byproduct we conclude that the volume of a finite-volume singular set vanishes at the singular time.We also define a notion of density for Type I Ricci flows and use it to prove a regularity theorem reminiscent of White's partial regularity result for mean curvature flow [22].

We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss-Bonnet with cutoff argument.

We consider a closed manifold M with a Riemannian metric gij (t) evolving by ∂t gij = −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 ∂t gij = −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 [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.

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