1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. SOME RESULTS ON TYPE I ANCIENT SOLUTIONS 161 Since f (-1) = f (-1), we conclude that }(t) = f (t). From (30.105) we conclude ...

162 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS By Corollary 30.11, the reduced distance functions ± C(~o,o) : M x (-oo, 0) - ...

SOME RESULTS ON TYPE I ANCIENT SOLUTIONS 163 PROPOSITION 30. 32 (Backwards limits with fixed basepoint of a shrinker are isome ...

164 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS Then by (30.109) we have so that \79^00 (tlm 00 (t) is a parallel gradient ve ...

4. SOME RESULTS ON TYPE I ANCIENT SOLUTIONS We conclude that for t < 0 (30.118) v~ (t) = r (- 47rt)-nf^2 e- et,(x,t)dμ,9oo(t) ...

166 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS Since M 00 is noncompact, by Theorem 9.66 in [77], there exists a sequence of ...

SOME RESULTS ON TYPE I ANCIENT SOLUTIONS PROOF. By (30.120), we have 1 2 2Rc(X) + V'xX + 2\7 IXI - X = 0. Thus ~x 1x1 2 = (\7 ...

168 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONS THEOREM 30.40 (Shrinkers with IRml ~ C must be gradient and K-non- collapsed) ...

NOTES AND COMMENTARY 169 Note the following: (a) A steady GRS may not necessarily be 11:-noncollapsed for any 11: > 0. For ...

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CHAPTER 31 Hyperbolic Geometry and 3-Manifolds We've got no future, we've got no past Here today, built to last. From "West End ...

172 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS A beautiful model of hyperbolic space is the hyperboloid model. Let !En,l denote Min ...

INTRODUCTION TO HYPERBOLIC SPACE 173 We leave it as an exercise for the reader to verify the lemma. HINT: Recall that an isome ...

174 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS ¢(Mi::ib(n-1)) is exactly the subgroup of Mi::ibius transformations of !Rn leaving u ...

l. INTRODUCTION TO HYPERBOLIC SPACE 175 An interesting property of inversions of JR.n is that they preserve the set of all ( cod ...

176 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS is a parabolic isometry. When n = 2, 3, every parabolic isometry is con- jugate to a ...

l. INTRODUCTION TO HYPERBOLIC SPACE 177 LEMMA 31.11 (Nilpotent subgroups of Isom (lH!n)). If G c Isom (lH!n) is a nilpotent subg ...

178 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS ( 2) Take the Riemann surface S = <C - { 0, 1}, the twice-punctured plane. Then S ...

2. TOPOLOGY AND GEOMETRY OF HYPERBOLIC 3 -MANIFOLDS 179 PROOF. Let 52 be an embedded 2-sphere in a complete hyperbolic manifold ...

180 31. HYPERBOLIC GEOMETRY AND^3 -MANIFOLDS respectively. By the classification of surfaces with boundary, there exists a homeo ...