1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

MONOTONICITY OF ENERGY FOR THE RICCI FLOW 197 From (5.13) we have (5.24) Ov? (g) =-JM Vij (Rij + \Ji\Jjf) dm, where f is given ...

198 5. ENERGY, MONOTONICITY, AND BREATHERS 2.1. A coupled system equivalent to the gradient flow of ;:m. There is a coupled syst ...

MONOTONICITY OF ENERGY FOR THE RICCI FLOW 199 2.2. Correspondence between solutions of the gradient fl.ow and solutions of the ...

200 5. ENERGY, MONOTONICITY, AND BREATHERS 2.2.2. Converting a solution of Ricci flow to a solution of the gradient flow. Now we ...

MONOTONICITY OF ENERGY FOR THE RICCI FLOW 201 since foT JM bOadμdt = foT JM b ( :t - ~) adμdt = foT JM [a ( -:t - ~) bdμ - ab ...

202 5. ENERGY, MONOTONICITY, AND BREATHERS D 2.3.2. Deriving the monotonicity of :F from a pointwise estimate. This second appro ...

STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED 203 On the other hand, Plugging in the equation for (gt+~) Rm and using (5.4 ...

204 5. ENERGY, MONOTONICITY, AND BREATHERS 3.1. The infimum A of :F. Suppose we have a steady breather solu- tion to the Ricci f ...

STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED 205 (1) the minimum value>.. (g) of 9(g, w) is equal to A.1(g), where > ...

206 5. ENERGY, MONOTONICITY, AND BREATHERS Finally we show wo > 0. By the Hopf boundary point lemma (see Lemma 3.4 of Gilbarg ...

STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED 207 (v) (Scaling) ).. (cg) = c -l).. (g). 3.2. The monotonicity of>.. Let ...

208 5. ENERGY, MONOTONICITY, AND BREATHERS so that JM w^2 dμ 91 Q(g2, w) - JM w^2 dμ 92 Q(g1, w) '.S 4 ( (1 + c;):g:+l - 1) JM w ...

STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED 209 The result now follows from 4 JM \Y'w\~1 dμg1 = 9(91, w) JM R 91 w^2 dμ ...

210 5. ENERGY, MONOTONICITY, AND BREATHERS SOLUTION TO EXERCISE 5 .27. We compute^13 _:!__ >. (g ( t)) I ~ lim inf >. (g ( ...

STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED 211 Note by (5.51) that f satisfies the equation 2b..f -1Vfl^2 + R = >. ( ...

212 5. ENERGY, MONOTONICITY, AND BREATHERS where f = f (t) is the minimizer of :F (g (t), ·). From this we obtain t v-^2 /n :t & ...

STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED 213 LEMMA 5.31. Expanding or steady breathers on closed manifolds are Einste ...

214 5. ENERGY, MONOTONICITY, AND BREATHERS Note that on a shrinking breather we have V (t2) < V (t1) for t2 > ti. In parti ...

CLASSICAL ENTROPY AND PERELMAN'S ENERGY 215 that is, T < n { dm. 2.rm(g(O)) JM The proposition is a consequence of the fol ...

216 5. ENERGY, MONOTONICITY, AND BREATHERS REMARK 5.37. (1) The formula above for ftFm is somewhat reminiscent of Hamilton's for ...