1547671870-The_Ricci_Flow__Chow

8 FINITE-TIME BLOWUP 209 and a sequence of times ti / T such that M (ti) ::; Ko. A consequence of the differential inequality :t ...

210 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE to the initial value problem { dr/dt = 2r^2 /n r(to) p By the maximum princip ...

FINITE-TIME BLOWUP By Theorem 6. 35 , there are positive constants A , B , and a such that IV R l 2 ::::: ~A 2 R~2°' + B 2 . 2 ...

212 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE Let x, y E M^3 and T/ > 0 be given. Then by (6.53), (6.55), and (6.56), th ...

PROPERTIES OF THE NORMALIZED RICCI FLOW 213 on a manifold of finite volume. To convert to the normalized fl.ow, define dilatin ...

214 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE This shows that ~ = -b ~t, hence by (6.57) that g evolves by the normalized R ...

PROPERTIES OF THE NORMALIZED RICCI FLOW 215 LEMMA 6.60. There exists a positive constant C such that PROOF. Let L (f) and V (f ...

216 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE R = R/1/J, and dp, = 1jJnl^2 dμ = 1jJ^312 dμ. Thus for any T E [O, T), we hav ...

PROPERTIES OF THE NORMALIZED RICCI FLOW 217 (Note that we are now dropping the .9 ([) notation used above.) The easiest way to ...

218 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE ( 4) The Ricci tensor of g evolves by a e at Rjk = b..LRjk = b..Rjk + 2RpjkqR ...

EXPONENTIAL CONVERGENCE 219 In particular, the eigenvalues>. ~ μ ~ v of M (which are twice the sectional curvatures) evolve ...

220 PROOF. and THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE Following the proof of Theorem 6.30, we compute that d - log ( ,\ - ...

NOTES AND COMMENTARY 221 for all positive time. From here it is not hard to show that all derivatives of the curvature decay exp ...

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CHAPTER 7 Derivative estimates We saw in Chapter 3 that the Ricci flow is equivalent via DeTurck's trick to a quasilinear parabo ...

224 then DERIVATIVE ESTIMATES IVm Rm (x, t)ig(x,t) :S ~:/~ a for all x E Mn and t E (0, K]. Note that the estimates in Theorem ...

l. GLOBAL ESTIMATES 225 Before proving Theorem 7.1, we will calculate the evolution of the square of the norm of the curvature t ...

226 7. DERIVATIVE ESTIMATES then M (t) ~ 2M (0) for all times 0 ~ t ~min { T , M~O)}. PROOF. By Lemma 7.4, M (t) is a Lipschitz ...

2. PROVING THE GLOBAL ESTIMATES 227 Simply put, in taking the time derivative of a quantity such as I 'V' Ql^2 , one must take i ...

228 7. DERIVATIVE ESTIMATES when we differentiate f\7 Rmf^2 by the good term -2(3 f\7 Rmf^2 we get when we differentiate fRmf^2. ...