1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

(^122) 19. GEOMETRIC PROPERTIES OF t;;-SOLUTIONS
As a special case we may take Vij = 2Rij, so that V = 2R, which implies
82Rc.C(/) = fo

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VT (~R + 2 JRcJ^2 - 2\7 R · '°Y + 2 Re ( ,-Y, ,-Y) - :) dr

• 2v'f R ("! (r), r).
  Suppose that the solution to the backward Ricci flow exists on the time
  interval [O, T] and r = T-t. Since Hamilton's trace Harnack estimate says

.6.R+ 2 JRcJ^2 - 2\7R · 1' + 2Rc (1',,-Y) + T ~

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2: 0,

we have

82Rc.C(i) 2: - fo

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JT (r ~ 7 + ~) R ("! (r), r) dr + 2v'fR (! (r) ,r).