By Fischer A.

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Anal. Geom. 7, 695–729. [27] Isenberg, J, and Jackson, M (1992), Ricci flow of locally homogeneous geometries on closed manifolds, J. Differential Geometry 35, 723-741. [28] Lang, S (1995), Differentiable and Riemannian manifolds, Springer-Verlag, New York. [29] Marsden, J, Ebin, D, and Fischer, A (1972), Diffeomorphism groups, hydrodynamics, and relativity, in Proceedings of the Thirteenth Biennial Seminar of the Canadian Mathematical Congress on Differential Topology; Differential Geometry and Applications, Volume 1, edited by J R Vanstone, Canadian Mathematical Society, Montreal, Canada, pp.

The current situation is summarized in Table 2. 9. The conformal Ricci flow and the σ-constant of M An important question regarding the conformal Ricci flow equation is under what conditions does a non-equilibrium initial condition g0 ∈ M−1 , Ric(g0 ) = − n1 g0 , have an all-time solution g : [0, ∞) −→ M−1 . 1) CONTENTS 45 Table 2. The volume and scalar curvature for non-static locally homogeneous solutions to the classical and conformal Ricci flow equation under either the assumption that R(gt ) < 0 or that M is of Yamabe type −1.

Define a rescaled reparameterized flow g¯ : [0, T¯) −→ M−1 , s −→ g¯(s) = |R(g(t(s))|g(t(s)) . 3) Then g¯(s) is a locally homogeneous solution to the conformal Ricci equations, ∂¯ g(s) + 2 Ric(¯ g (s)) + n1 g = − p(s)¯ g (s) ∂s R(¯ g(s)) = − 1 , with initial value g¯0 = |R(g0 )|g0 ∈ M−1 , with spatially constant conformal pressure p(s) = 2 |RicT(¯ g (s))|2g¯(s) , and if g is non-static, with strictly monotonically decreasing volume vol(M, g¯(s)) = |R(g(t(s))|n/2 . 4) Proof: Since the flow t → gt is locally homogeneous, the scalar curvatures are ¯ t) = constant, R(gt ) = ct = constant.