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Monotone volume formulas for geometric flows, arXiv:0905.2328
"... We consider a closed manifold M with a Riemannian metric gij(t) evolving by ∂t gij = −2Sij where Sij(t) is a symmetric twotensor 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 construc ..."
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We consider a closed manifold M with a Riemannian metric gij(t) evolving by ∂t gij = −2Sij where Sij(t) is a symmetric twotensor 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 nonincreasing, the latter being nondecreasing along the flow ∂t gij = −2Sij. In the case where Sij = Rij, the Ricci curvature of M, the result corresponds to Perelman’s wellknown 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. 1 Introduction and formulation of the main result Let M be a closed manifold with a timedependent Riemannian metric gij(t). Let S(t) be
Ricci Solitons and EinsteinScalar Field Theory
, 2009
"... B List has recently studied a geometric flow whose fixed points correspond to static Ricci flat spacetimes. It is now known that this flow is in fact Ricci flow modulo pullback by a certain diffeomorphism. We use this observation to associate to each static Ricci flat spacetime a local Ricci soliton ..."
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B List has recently studied a geometric flow whose fixed points correspond to static Ricci flat spacetimes. It is now known that this flow is in fact Ricci flow modulo pullback by a certain diffeomorphism. We use this observation to associate to each static Ricci flat spacetime a local Ricci soliton in one higher dimension. As well, solutions of Euclideansignature Einstein gravity coupled to a free massless scalar field with nonzero cosmological constant are associated to shrinking or expanding Ricci solitons. We exhibit examples, including an explicit family of complete expanding solitons which can be thought of as a Ricci flow for a complete Lorentzian metric. The possible generalization to Ricciflat stationary metrics leads us to consider an alternative to Ricci flow.
Harnack estimates for ricci flow on a warped product. preprint arXiv:1211.6448
, 2012
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Ricci flow on threedimensional manifolds with symmetry
 Comment. Math. Helv
, 2014
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Static flow on complete noncompact manifolds i: shorttime existence and asymptotic expansions at conformal infinity
 Science China Mathematics
, 2012
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