1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 321 on Bg(T) (p, R), where g is some complete metric on M and Co < ...

322 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS 2.3. The Harnack quantity. We proceed to give the proofs of Theorem ...

LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 323 where L = log u, satisfies the heat-type equation (25.67) aP 87 = ...

324 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS PROOF. Applying the assumed bounds in (25.69) to the evolution equa- ...

LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 325 point where ¢ =I= 0 ~!!__ (T</>P) = ¢ oP + 0¢ p +</JP T ...

326 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS wherever P < 0, we obtain from (25.75) that (25. 78) 0;::: 2-3c & ...

LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 327 Since (xo, To) is a point where T</> (x) P (x, T) attains a ...

328 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS To obtain upper bounds for 1v:1 2 and ~~ - bi.¢, evidently it suffic ...

LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 329 and (2) (uniform bounds on its first and second covariant derivat ...

330 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS since ~~ = 0 and by (25.95) and (25.94). By our definition of C3 in ...

NOTES AND COMMENTARY 331 in x E M - Bg(r)(P, }g) for each T E [O, T]. Since R 2:: Jg and for each T E [O, T] we have¢= 1 in Bg ...

332 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS original papers [135] and [134] for both the parabolic and elliptic ...

Chapter 26. Bounds for the Heat Kernel for Evolving Metrics Only love can bring the rain. From "Love, Reign o'er Me" by The Who ...

334 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS We consider the heat(-type) equation (26.4) OU Lx,TU ~ or - llg(T)U + Qu ...

HEAT KERNEL FOR AN EVOLVING METRIC 335 A and B are functions on M x [O, T] which are both C^2 in space and C^1 in time, then f ...

336 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS in (26.11). Since (LxH) (x,7;z,p) = 0 and (L;H*) (x,7;y,v) = 0, we have ...

HEAT KERNEL FOR AN EVOLVING METRIC we have Id~ (JM H (x, r; y, v) dμ 9 ( 7 ) (x)) I ::S C1 JM H (x, r; y, v) dμ 9 ( 7 ) (x). O ...

338 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS Given y, z E M and 0 ::; v < p ::; T, using (26.6) and (26.15), we co ...

HEAT KERNEL FOR AN EVOLVING METRIC 339 In particular, (26.25) (a:: + ,6,,y,vHD) (x, Ti y, v) + (R - Q) (y, v) HD (x, Ti y, v) ...

340 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS where v denotes the unit outward normal vector field to M on 8M, where V ...