By Fischer A.

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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). Thus a graph manifold is a union of Seifert fibered spaces glued together by toral automorphisms along toral boundary components.

Similarly, since for all h ∈ S2 , M M M (∆g trg h + δg δg h − Ric(g) · h) + = M = M R(g) D(dµg )h DR(g)h dµg + D(R(g)dµg )h = (dRtotal )(g)h = 1 2 R(g) trg h dµg M (−Ric(g) + 21 R(g)g) · h dµg = − Ein(g) · h dµg = M (grad Rtotal )(g) · h dµg , M we have (grad Rtotal )(g) = −Ein(g). With these two basic gradients in hand, we find (grad Y )(g) = (grad vol(2−n)/n Rtotal )(g) = (grad vol(2−n)/n )(g)Rtotal (g) + vg(2−n)/n (grad Rtotal )(g) = = 2−n (2−n)/n −1 1 )vg ( 2 g)Rtotal (g) − n (vg (2−n)/n ¯ Ein(g) + n−2 − vg 2n R(g)g vg(2−n)/n Ein(g) ¯ = − vg(2−n)/n (Ric(g) − 21 R(g)g) + ( 12 − n1 )R(g)g ¯ ¯ = − vg(2−n)/n (Ric(g) − n1 R(g)g) + 12 (R(g) − R(g))g .

15), we can formulate the conformal Ricci flow equations in a gradient-like manner, which we refer to as a quasi-gradient. 1 (A quasi-gradient form of the conformal Ricci flow equations) The gradient of the Yamabe functional ¯ Y : M −→ R , g −→ vg(2−n)/n Rtotal (g) = vg2/n R(g) in the natural L2 -Riemannian metric G on M is given by grad Y : M −→ S2 , g −→ (grad Y )(g) = −vg(2−n)/n Ein(g) + = −vg(2−n)/n n−2 ¯ 2n R(g)g ¯ ¯ + 21 (R(g) − R(g) g . Ric(g) − n1 R(g)g When restricted to M−1 , grad Y simplifies to (grad Y )|M−1 : M−1 −→ S2 , g −→ (grad Y )|M−1 (g) = −vg(2−n)/n Ric(g) + n1 g = −vg(2−n)/n RicT(g) .

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