1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

VOLUME GROWTH OF SHRINKING GRADIENT RICCI SOLITONS 21 Since the !-Ricci tensor is nonnegative, i.e., Re / = Re+ \7^2 f = ~ g 2 ...

22 27. NONCOMPACT GRADIENT RICCI SOLITONS COROLLARY 27.36 (Volume growth of shrinkers with R :'.'.: o :'.'.: 0). If Q = (Mn, g , ...

VOLUME GROWTH OF SHRINKING GRADIENT R ICCI SOLITONS 23 together, we obtain the !-Bochner formula (27.96) 1 2.6.1l'Vul^2 = l\7^ ...

24 27. NONCOMPACT GRADIENT RICCI SOLITONS since lim,._, 0 rn-l J 1 (r) = e-f(Ol. Integrating this yields the following (one deal ...

VOLUME GROWTH OF SHRINKING GRADIENT RICCI SOLITONS 25 By combining (27.107) and (27.108), we obtain (27.110) ~ (nn J(f)) = ln ...

26 27. NONCOMPACT GRADIENT RICCI SOLITONS In general, for shrinkers, although limr-+oo Vol~no(r) exists for each 0 EM and is bou ...

5. LOGARITHMIC SOBOLEV INEQUALITY 27 REMARK 27.47. In comparison, Corollary 6.38 in Part I is equivalent to the following statem ...

28 27. NONCOMPACT GRADIENT RICCI SOLITONS Therefore, by (27.125) we have :tH(~(t)) :'.'.'. :tl(~(t)). Under suitable growth cond ...

6. GRADIENT SHRINKERS WITH NONNEGATIVE RICCI CURVATURE 29 By Perelman's proof of his no loca l collapsing theorem, one can demon ...

30 27. NONCOMPACT GRADIENT RICCI SOLITONS and hence (27.134) R(xo) 2 R(u(u)) for all u E (-oo,O]. Claim. For any x 0 EM - Ba(Sr ...

GRADIENT SHRINKERS WITH NONNEGATIVE RICCI CURVATURE 31 for any continuous piecewise c= function ( : [O, r (x)] -t JR satisfyin ...

32 27. NONCOMPACT GRADIENT RICCI SOLITONS Therefore (27.144) 1 ;~~~ro R ('y(s)) ds::::; e4(n-1)(1+2v'J(O))ro. Combining this wit ...

N OTES AND COMMENTARY 33 N ates and commentary There are m any works on GRS which we have not discussed in this chapter. We hav ...

34 27. NONCOMPACT GRADIENT RICCI SOLITONS §5. Theorem 27.46 is in [53]. Their proof is based on the earlier work in [16] and [42 ...

Chapter 28. Special Ancient Solutions When you're ridin' sixteen hours and the re's nothin' 1nuch to do A n d you don't feel muc ...

36 28. SPECIAL ANCIENT SOLUTIONS STEP l. The cutoff function. First we improve the standard cutoff function slightly. Given p E ...

LOCAL ESTIMATE FOR THE SCALAR CURVATURE UNDER RICCI FLOW 37 so that (28.10) ( f) (8R ) r/ o i. (or ) = 'T/ o p 7ft - 6.R + R-; ...

38 28. SPECIAL ANCIENT SOLUTIONS STEP 5. ODI comparison gives a lower bound for Smin· Recall that in general if we have a soluti ...

l. LOCAL ESTIMATE FOR THE SCALAR CURVATURE UNDER R ICCI FLOW 39 COROLLARY 28.2 (Ancient solut ions h ave nonnegative scalar curv ...

40 28. SPECIAL ANCIENT SOLUTIONS Regarding t he sectional curvatures of a ncient solutions, B .-L. Chen h as proved the followin ...