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4) Proof: Since the flow t → gt is locally homogeneous, the scalar curvatures are ¯ t) = constant, R(gt ) = ct = constant. 5) ∂t with R(g) spatially constant. 6) n ∆g R(g) + 2|Ric(g)|g − n R (g) = 2|Ric(g)|g − n R (g) , ∂t which is no longer a backwards heat equation. 17) is equal to the classical Ricci flow equation and is well-posed (see also the remark right after this proof). Also, d dµg = 21 trg dt ∂g ∂t dµg = 21 trg −2 Ric(g) + n2 R(g)g dµg = (−R(g) + R(g))dµg = 0 , so that the volume element (and not just the volume) of the flow is preserved.

Now assume that the spatially constant scalar curvatures satisfy R(gt ) = ct < 0. First we rescale and then we reparameterize the classical Ricci solution g : [0, T ) → M1 . Since for each t ∈ [0, T ), R(gt ) = ct < 0 is constant, we can rescale gt by a time-dependent homothetic transformation to get g˜t = |R(gt )|gt = −R(gt )gt . 7) R(˜ gt ) = R(|R(gt )|gt ) = |R(gt )| so that g˜t ∈ M−1 satisfies the constraint equation of the conformal Ricci system. 6) that g˜ satisfies the evolution equation ∂˜ g ∂R(g) ∂g = − g − R(g) ∂t ∂t ∂t = − 2|Ric(g)|2g − n2 R2 (g) g − R(g) − 2 Ric(g) + n2 R(g)g = − 2|Ric(g)|2g g + 2R(g) Ric(g) = − 2|R(g)|2 |Ric(˜ g )|2g˜ g − 2|R(g)| Ric(˜ g) = − 2|R(g)||Ric(˜ g )|2g˜ g˜ − 2|R(g)| Ric(˜ g) .

12) n 2 s + 1, R : M → F DR(g)(ϕg) = (n − 1)∆g ϕ − R(g)ϕ = Lg ϕ = 0 . 13) has unique solution ϕ = 0. To show that the direct sum S¯2s ⊕ F s g of the indicated closed subspaces exhausts s s−2 S2 , for h ∈ S2s let ϕ = L−1 , by the ellipticity of Lg , g (DR(g)h). 9). 9) does since it is required that g is such that Lg = (n − 1)∆g − R(g) is an isomorphism. 11) of S2s can be interpreted geometrically as a splitting of the tangent space Tg Ms into the tangent CONTENTS 31 spaces of two closed transversally intersecting submanifolds of Ms , namely, Msρ and P s g.

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