1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

CHAPTER 27 Noncompact Gradient Ricci Solitons There is no elite Just take your place in the drive r's seat. Fron1 "Driver's Sea ...

2 27. NONCOMPACT GRADIENT RICCI SOLITONS application of the maximum principle to the elliptic equation satisfied by the scalar c ...

BASIC PROPERTIES OF GRADIENT RICCI SOLITONS 3 Define the !-Ricci tensor to be Re t ~ Re + \7^2 f; this is also known as the Ba ...

4 27. NONCOMPACT GRADIENT RICCI SOLITONS for some universal const < oo. We define <Pc : M ~JR by (27.12) <Pc (x)=rJ (r( ...

l. BASIC PROPERTIES OF GRADIENT RICCI SOLITONS 5 for any unit speed minimal geodesic 'Y : [O, r (xe)] ---+ M joining 6 to Xe and ...

6 27. NONCOMPACT GRADIENT RICCI SOLITONS STEP 4. Proof of the theorem when c: = l. Define 'T/ : [O, oo) ~ [O, 1] so as to satisf ...

l. BASIC PROPERTIES OF GRADIENT RICCI SOLITONS 7 THEOREM 27.4 (Bounds for IV'fl). Suppose that (Mn,g, f,c) is a complete normali ...

8 27. NONCOMPACT GRADIENT RICCI SOLITONS Combining Theorem 27.4 with Lemma 27.6 yields the following result of Z.-H. Zhang. CORO ...

ESTIMATES FOR POTENTIAL FUNCTIONS OF GRADIENT SOLITONS 9 Thus f is uniformly convex and proper. In particular, f attains its i ...

10 27. NONCOMPACT GRADIENT RICCI SOLITONS COROLLARY 27.10 (Bounds on f for GRS). Let (Mn,g, f , c) be a complete normalized G RS ...

ESTIMATES FOR POTENTIAL FUNCTIONS OF GRADIENT SOLITONS 11 PROOF. Let x EM - B 0 (2) be any point and let 1: [O, r (x)] --+ M b ...

12 2 7. NONCOMPACT GRADIENT RICCI SOLITONS PROOF. It follows from (27.47) that 35 d(x, 01 ) ::; 2Vf (x) + 8 n for all x EM. Taki ...

ESTIMATES FOR POTENTIAL FUNCTIONS OF GRADIENT SOLITONS 13 Since the assumption (27.54) is on the edge of holding true in the s ...

14 27. NONCOMPACT GRADIENT RICCI SOLITONS Since 2 J;(~/-li r(~~-s ds = 1, there exists s E [r (x) - cl, r (x)] such that y ~ 1(8 ...

LOWER BOUNDS FOR R OF NONFLAT NONEXPANDERS 15 We have the following results of Y. Deng and X. Zhu. THEOREM 27.24. Let Q = (M, ...

16 2 7. NONCOMPACT G R ADIENT RICCI SOLIT O N S By (27.42) and (27.46), we h ave good estimat es on the potent ial function: (27 ...

VOLUME GROWTH OF SHRINKING GRADIENT RICCI SOLITONS 17 By this and (27.10), we obtain for c E JR, 2 nc^2 1 (R - eel) ::::; --; ...

18 27. NONCOMPACT GRADIENT RICCI SOLITONS Henceforth, let (Mn, g, f, -1) be a complete normalized noncompact shrinking GRS struc ...

VOLUME GROWTH OF SHRINKING GRADIENT RICCI SOLITONS 19 (here we recall that by (27.46), f is a proper function). Hence we verif ...

20 27. NONCOMPACT GRADIENT RICCI SOLITONS THEOREM 27.33 (The AVR exists and is bounded for shrinkers). Let (Mn,g, f , -1) be a c ...