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By Fischer A.

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The main idea is to try to show that the Ricci flow evolves any given initial Riemannian manifold (M, g0 ) to a geometric structure after performing a finite number of surgeries as curvature CONTENTS 47 singularities arise in finite time during the Ricci flow. The hope is that if the surgeries are done before the singularities arise, the Ricci flow can be continued for all time and the initial manifold (M, g0 ) will naturally decompose into geometrical pieces. In this picture the classical Ricci flow causes the initial Riemannian manifold (M, g0 ) to selfgeometrize to a limit space (M∞ , g∞ ) which is the natural geometrical decomposition of (M, g0 ).

Differential Geom. 20, 479–495. [40] Taylor, M (1996) Partial Differential Equations III, Nonlinear Equations, Springer, New York. [41] Thurston, W (1997), Three-dimensional geometry and topology, volume 1, edited by S Levy, Princeton University Press, Princeton, New Jersey. CONTENTS 50 [42] Wheeler, J A (1962), Geometrodynamics, Academic Press, New York. [43] Wheeler, J A (1968), Superspace and the nature of quantum geometrodynamics, in Battelle Rencontres – 1967 Lectures in Mathematics and Physics, C.

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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