1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

CLASSICAL ENTROPY AND PERELMAN'S ENERGY 217 where we integrated by parts and used Holder's inequality and the Gauss- Bonnet fo ...

218 5. ENERGY, MONOTONICITY, AND BREATHERS 4.2.3. The gradient of Hamilton's surface entropy is the matrix Harnack quantity. A l ...

NOTES AND COMMENTARY 219 By limt->oo <lJf (t) = 0 and (5.71), we have dE ()Q d^2 E f 00 dE dt (0) = - lo dt2 (t) dt::::: ...

220 5. ENERGY, MONOTONICITY, AND BREATHERS using ~ = h ~ 8 f. Integrating by parts yields 8:F = 8 JM Re-f dμ + 5 JM IVJl^2 e-f d ...

Chapter 6. Entropy and No Local Collapsing Everything should be made as simple as possible, but not simpler. Albert Einstein Di ...

222 6. ENTROPY AND NO LOCAL COLLAPSING >.(g(t)) < 0 for some t E [t1, t2], even shrinking breathers, i.e., solutions with ...

THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY 223 1.1.2. The first variation of W. Let Og = v E C^2 (M, T M @T M), let of= h, ...

224 6. ENTROPY AND NO LOCAL COLLAPSING Combining the above three formulas and simplifying a little, we get b(v,h,() W (g, f, r) ...

THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY 225 i.e., the variation preserves the measure (47rT)-nl^2 e-f dμ on M, we obtain ...

226 6. ENTROPY AND NO LOCAL COLLAPSING THEOREM 6.4 (Entropy monotonicity for Ricci fl.ow). Let (g(t),J(t),r(t)), t E [O, T], be ...

THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY 227 to (5.31), so that we get d:F r 2 -! n dt = 2 JM IRij + Y'iY'jfl e dμ - 27 F ...

228 6. ENTROPY AND NO LOCAL COLLAPSING Theorem 6.4 then follows from dW = f (~ - R) vdμ = { (-D* - .6.) vdμ dt JM at JM = 2T JM ...

THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY 229 that since D* [ (logu + ~ log(47rT) + n) u] = uD* log u +log uD*u - 2 ('V lo ...

230 6. ENTROPY AND NO LOCAL COLLAPSING where u is defined in (6.3). When c: < 0, this is Perelman's entropy; when c: = 0, thi ...

THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY 231 EXERCISE 6.13 (First variation formula for We:). Show that, on a closed mani ...

232 6. ENTROPY AND NO LOCAL COLLAPSING Since Wc:(g, f, T) = JM (T (R + .6.f) - c(J - n)) (47rT)-nl^2 e-f dμ, inte- grating this ...

THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY 233 EXERCISE 6.15 (Evolution of v 6 and monotonicity of W 6 ). Consider the gaug ...

234 6. ENTROPY AND NO LOCAL COLLAPSING EXERCISE 6 .17. Let (6.41) so that w^2 = u. Show that (6.42) w ( f ) = r ( T ( Rw 2 + 4 I ...

THE FUNCTIONALS μ AND v 235 of this chapter). In particular, by taking a = l~ in (6.65) below, we have for£< 0, WE:(g, f, T ...

236 6. ENTROPY AND NO LOCAL COLLAPSING DEFINITION 6.20 (Infimum invariantsμ (g, T) and v (g)). The function- alsμ : 9J1et x JR+ ...