1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

SOME FURTHER CALCULATIONS RELATED TO :F AND W 277 Differentiating (5.10) again and using (6.106) and (6.107), we have d2 ds 2 ...

278 6. ENTROPY AND NO LOCAL COLLAPSING If g is a steady gradient soliton flowing along 'V f, then Rij + 'Vi'Vjf = 0, 2Af-J'VJJ^2 ...

SOME FURTHER CALCULATIONS RELATED TO :F AND W 279 If a is a 1-form, then the minimizer of the energy E(h) ~JM ldh+al^2 dμ is g ...

280 6. ENTROPY AND NO LOCAL COLLAPSING REMARK 6.92. Note that (V'i - V'd) ('Yj - Y'jf) Vij =div (divv) +\Re, v) - 2 div (v) · V' ...

SOME FURTHER CALCULATIONS RELATED TO :F AND W 281 and let v be a symmetric 2-tensor. If g (s) = g +sh, then d 2 -d 2 I v (g (s ...

282 6. ENTROPY AND NO LOCAL COLLAPSING for any function¢ of space and time. We may also rewrite the evolution of Re in the follo ...

SOME FURTHER CALCULATIONS RELATED TO :F AND W 283 EXERCISE 6.98. Under the system of equations 8 at9ij = -2Rij, 8 at X = -LlX ...

284 6. ENTROPY AND NO LOCAL COLLAPSING 8. Notes and commentary Section 5. For T < oo the condition on rVtk just says that the ...

Chapter 7. The Reduced Distance How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand ...

286 7. THE REDUCED DISTANCE /'\;-solutions, finite-time singularity models, and high curvature re- gions of the solution. ( 5) R ...

THE .C-LENGTH AND DISTANCE FOR A STATIC METRIC 287 solution to the heat equation, which is a delta function at 7 = 0 at a poin ...

288 7. THE REDUCED DISTANCE EXERCISE 7.2 (£-distance and .e between two space-time points). Show using (7.1) that inf 7 £ 71 , 7 ...

THE £-LENGTH AND THE £-DISTANCE 289 by reversing time. In particular, if w < +oo, let r ~ w-t, so that (N, h (r)) is a solu ...

290 7. THE REDUCED DISTANCE Lengthri (;y) = fo'Tq J ~ + R + l'Y (7)J^2 d7 = lrq [f;J 1 + ~ ( R + J')t (7)J^2 )d7 = rq (N (l+N 7 ...

THE £-LENGTH AND THE £-DISTANCE 291 In particular, Nf'~b are independent of N, whereas hm · N-o r .. = 0 N-+oo iJ ' hm · N-o r ...

(^292) 7. THE REDUCED DISTANCE above approach of renormalizing the Riemannian length functional, is as follows. We define the sp ...

THE £-LENGTH AND THE £-DISTANCE 293 2.3. The £-distance function. Just as for the usual length functional (perhaps it is bette ...

294 7. THE REDUCED DISTANCE where f3 : [O, 2v'TJ ~ Nn is a minimal constant speed geodesic with respect to ho joining p to q. Th ...

THE £-LENGTH AND THE L-DISTANCE 295 where f3 (CT)~ I (r), CT= 2..,/T, and CT*~ 2~. (iii) (Bounding L by Riemannian distance) F ...

296 7. THE REDUCED DISTANCE By the mean value theorem for integrals, there exists T* E (0, f) such that (iii) Let 'fl : [o, 2-JT ...