1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

APPROXIMATE ISOMETRIES, COMPACTNESS OF MAPS, DIRECT LIMITS 157 where x.rv y if x E Ak and y E Ae for some k,£ EN and either fk ...

158 4. PROOF OF THE COMPACTNESS THEOREM COROLLARY 4.16 (Second-countable direct limits). If each Ak is the countable union of co ...

CONSTRUCTION OF GOOD COVERINGS BY BALLS 159 Bk C (Mk, gk), and functions A (r), K (r), I (n, Co), where Co is the cur- vature ...

160 4. PROOF OF THE COMPACTNESS THEOREM 3.2. Choice of ball centers. In this subsection we will find centers for the balls which ...

CONSTRUCTION OF GOOD COVERINGS BY BALLS 161 PROOF. There are two steps to the argument. The first is that the balls { B ( xa, ...

162 4. PROOF OF THE COMPACTNESS THEOREM In summary we have constructed a sequence of balls such that the fol- lowing holds: PROP ...

CONSTRUCTION OF GOOD COVERINGS BY BALLS 163 where c and Care defined by .A [rJ ~ ce-Cr as in (4.7), i.e., (4.8) a c ~ D · min{ ...

164 4. PROOF OF THE COMPACTNESS THEOREM such that K (ae, f3e) E Ke is a monotone function of£. If k 2: K (ae, f3e), then k E Kg. ...

THE LIMIT MANIFOLD (M~,9 00 ) 165 Finally, we want to ensure that the Bk are embedded geodesic balls and to ensure that the re ...

166 4. PROOF OF THE COMPACTNESS THEOREM as restrictions of the map expx"' k oLf., where Ea, JJJa, and iJa are the appropriately ...

THE LIMIT MANIFOLD (M~,g 00 ) 167 To obtain the local convergence of Jff3 as k -+ oo, we take some fur- ther subsequences. Sin ...

168 4. PROOF OF THE COMPACTNESS THEOREM Hence we have the following local property which is a key step to Proposition 4.33. PROP ...

THE LIMIT MANIFOLD (M~,g 00 ) 169 Notice that with respect to the coordinates ( Ef3, Hf) the map q/{; ( x) , where a i= 0, can ...

170 4. PROOF OF THE COMPACTNESS THEOREM Define the local versions of Fk£;r by Gf P.;r ~ ( fif)- 1 o FkP.;r o Hf. We may pull bac ...

THE LIMIT MANIFOLD (M~,g 00 ) 171 where I· I is the Euclidean norm and '\7 is the Euclidean covariant derivative {i.e., partia ...

172 4. PROOF OF THE COMPACTNESS THEOREM Next we turn to prove that Fk£;r is a diffeomorphism. Since F[k is the inverse of Ffj, b ...

THE LIMIT MANIFOLD (M;;,,,g 00 ) 173 Similarly, we have again since Cr 2: 1. Hence, given E > 0 and p > 0, we can take j ...

174 4. PROOF OF THE COMPACTNESS THEOREM 4.4. Construction of the limit. We are now ready to construct the limit manifold (M~, g ...

CENTER OF MASS AND NONLINEAR AVERAGES 175 implies J/;^1 (13) is closed and bounded, and hence compact since 9k is com- plete. ...

176 4. PROOF OF THE COMPACTNESS THEOREM where we are considering the curves Is both in terms of geodesics from x and geodesics f ...