1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

CLASSIFY ING THE BACKWARD POINTWISE LIMIT 101 circular averages as well as on a concentration-compactness-type result which im ...

102 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS 9.2. A concentration-compactness result. The following is a concentration-compac ...

CLASSIFYING THE BACKWARD POINTWISE LIMIT 103 Now let z = w - w. Then lieucZ = 0 in Bz 0 (r). Since z+ is subha rmonic on B z 0 ...

104 29. COMPACT 2 -DIMENSIONAL ANCIENT SOLUTIONS 9.4. Proof of the classification of v 00 • We now classify the backward pointwi ...

CLASSIFYING THE BACKWARD POINTWISE LIMIT 105 Thus This completes the proof of Claim 1. Now, by Corollary 6.3 in [194], (29.110 ...

106 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS 10. An unrescaled cigar backward Cheeger- Gromov limit Let (5^2 , g(t)), t E (-o ...

10. AN UNRESCALED CIGAR BACKWARD CHEEGER-GROMOV LIMIT 107 at the point N and using the diffeomorphisms Wi ~ cr-^1 o <I>i o ...

108 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS such that Vi converges uniformly to a C^00 function v 00 : IR^2 x ( -oo, oo) --t ...

IRREDUCIBLE COMPONENTS OF 'i7^3 v 109 Since IAl 2 = ~ IV'vl 2 and /3 \ 23 3 2 2 \ V' v, A 1 = - 3 ('Vijkv - 'Vjikv)'Vjv(gs2 )i ...

110 29. COMPACT 2 -DIMENSIONAL ANCIENT SOLUTIONS The 3-t ensor H is fully symmetric and trace-free and is easily seen to b e H = ...

12 The heat-type equation satisfied by Q EXERCISE 29.43. (1) Show t hat S (V^3 vKR) = 4/3 sin 2'lj; S (d'l/; 0 g5 2 ). From this ...

112 29. COMPACT^2 -DIMENSIONAL ANCIENT SOLUTIONS The first term on the RHS is equal to 'Vi.6. 52 v;kv = v rnjkv = \7 Wj^5 k V - ...

THE HEAT-TYPE EQUATION SATISFIED BY Q 113 Toward the goal of obtaining nonpositive terms on the RI-IS of (29.147), we first re ...

114 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS Applying (29.148), (29.149), (29.152), (29.153), andRv = (vl:. 52 v-IV'vl^2 +2v^ ...

THE HEAT-TYPE EQUATION SATISFIED BY Q 115 where Z = ~ \7 6. 5 2v + idv. Now 1 2. 3tr' (\7 S(Z 0 g52))jk = \7' (Zi(gs2)jk + Zj( ...

116 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS Third, regarding the third line of equation (29.154), observe that by using that ...

13. THAT Q = 0 IMPLIES THE SOLUTION IS THE KING-ROSENAU SOLUTION 117 We also compute TF(B) ·TFS (V^3 v 83,2 V^2 v) = TF(B) · V^3 ...

118 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS To define the plane version Q of Q, we first consider the irreducible orthogonal ...

13. THAT Q = 0 IMPLIES THE SOLUTION IS THE KING-ROSENAU SOLUTION 119 We also have that V1112 = 3v1222 and V2221 = 3v1121, which ...

120 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS By Proposition 29.36 and by assuming without loss of generality that μ = 1, we h ...