1547671870-The_Ricci_Flow__Chow

7. STRATEGY FOR THE CASE THAT x (M^2 > 0) 129

are the metrics of constant curvature. Taken together, these observations
suggest one might be able to show that g (t) converges to a metric of constant
positive curvature by proving that M decays sufficiently rapidly. To explore
this idea, we compute the evolution equation satisfied by M.

LEMMA 5.34. On a solution (M^2 ,g(t)) of the normalized Ricci flow,

the tensor M evolves by

[)

at M = !:'::.M + (r - 2R) M.

PROOF. By Lemma 3.2, the Levi-Civita connection of g evolves by

Using this formula and recalling equations (5.3) and (5.10), we calculate

that

Next we use the fact that