By Fischer A.

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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. 15) ˜ as the tangential denote the projection onto S¯2s (g) = ker DR(g) and refer to h component (or part) of h and ϕg as the (infinitesimal) conformal component (or part) of h.

However, the theorem gives only a sufficient condition to prevent volume collapse. Thus, for example, a conformal Ricci flow g → ge that converges at any rate as t → ∞ to an Einstein metric ge ∈ M−1 , Ric(ge ) = − n1 ge , has vol(M, g) → vol(M, ge ) > 0 as t → ∞ and so the volume does not collapse. Similarly, if A(t) = maxx∈M |RicT(g(t, x))|2g(t,x) = A = constant, or more generally, if A(t) ≥ ǫ > 0 is is bounded away from zero, then the hypothesis of the theorem is not met and so, as far as the theorem goes, the volume may or may not collapse.

Under these circumstances, g cannot converge to a limit metric g∞ ∈ M−1 for if it did, by the continuity of the volume functional, vol(M, g) → vol(M, g∞ ) > 0, contradicting vol(M, g) → inf vol−1 = 0. Thus the flow must degenerate and the nature of the degeneration is of importance. , a space that admits a foliation by circles with the property that a foliated tubular neighborhood D2 × S 1 of each leaf is either the trivial foliation of a solid torus D2 × S 1 or its quotient by a standard action of a cyclic group (see Anderson [1] for more information about graph manifolds).