1547671870-The_Ricci_Flow__Chow

STRATEGY FOR THE CASE THAT x (M^2 > 0) 129 are the metrics of constant curvature. Taken together, these observations sugges ...

130 5. THE RICCI FLOW ON SURFACES on a surface to compute the commutator [V'V', ~],obtaining \i'i\i'j~f = \i'i\i'j\i'k\i'kf = \7 ...

STRATEGY FOR THE CASE THAT x (M^2 > 0) 131 As we saw in Section 4 of Chapter 2, the solution to (5.23) differs from the sol ...

132 5. THE RICCI FLOW ON SURFACES curvature allows us to apply Klingenberg's Theorem to obtain an injectivity radius bound. STEP ...

SURFACE ENTROPY 133 Surface entropy Since we have not been able to employ the maximum principle to obtain a uniform upper boun ...

134 5. THE RICCI FLOW ON SURFACES PROPOSITION 5.39. If (M^2 , g (t)) is a solution of the normalized Ricci flow on a compact sur ...

SURFACE ENTROPY 135 PROOF. If dN/dt = 0 at some time to E [O, oo), then M(·,to) = 0. By equation (5.7), g (to) is a gradient R ...

136 5. THE RICCI FLOW ON SURFACES When the initial metric has strictly positive curvature, Corollary 5. 40 shows that the entrop ...

UNIFORM UPPER BOUNDS FOR R AND IV RI PROOF. By (5.10) and (5.2), Hence :t JM 2 f dA = JM 2 ((!:lf + rf) + f (r - R)) dA = J (( ...

138 5. THE RICCI FLOW ON SURFACES 9.1. Bounds for the metric on a surface with x (M^2 ) > 0. Define Rmin (t) ~ minxEM2 R (x, ...

UNIFORM UPPER BOUNDS FOR R AND l'V RI 139 If to ::::; t ::::; to+ 1/2Rmax (to), then Lemma 5.45 implies that l t (r - R(x,T)) ...

140 5. THE RICCI FLOW ON SURFACES When the initial scalar curvature is nonnegative, these two lemmas lead to the following obser ...

9 UNIFORM UPPER BOUNDS FOR R AND l'V RI 141 at a minimum of R, we have Rmin (t) ~ min {O, Rmin (0)} ~ -K,. And since r ~ 0, we k ...

142 5. THE RICCI FLOW ON SURFACES Let to ~ t 1 - l"' > 0. Examining the proof of Proposition 5.50, we see that if JR J:::;: " ...

DIFFERENTIAL HARNACK ESTIMATES OF LYH TYPE 143 The uniform upper bound for the scalar curvature implies a uniform upper bound ...

144 5. THE RICCI FLOW ON SURFACES 10.1. The case that R (-, 0) > O. Recall that the gradient Ricci soliton equation ( 5. 7) i ...

DIFFERENTIAL HARNACK ESTIMATES OF LYH TYPE 145 COROLLARY 5.56. Q satisfies the evolutionary inequality a (5.38) at Q 2 b.Q + 2 ...

146 5. THE RICCI FLOW ON SURFACES where the infimum is taken over all C^1 - paths ry : [t 1 , t2] ---+ M^2 joining x1 and x2. Th ...

DIFFERENTIAL HARNACK ESTIMATES OF LYH TYPE 147 PROOF. As in Lemma 5.55, we compute at fj Q A = ( R - r) b.L A + b. ( b.L A + I ...

148 5. THE RICCI FLOW ON SURFACES and hence reach the conclusion. D This is our differential Harnack inequality in the case that ...