1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

CRITICAL POINT THEORY AND PROPERTIES OF DISTANCE SPHERES 481 We define the function f : 1-l n V -+ IR by (I.40) <]? (y) -:- ...

482 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION is a smooth invertible map. By originally choosing U sufficiently small (in ...

APPROXIMATE BUSEMANN-FELLER THEOREM 483 (2) for any y E 1-l with d (x, y) = d (x, 1-l) and for any unit speed minimal geodesic ...

484 I. ASYMPTOTIC CONES AND SHARAFUTDINOV .RETRACTION (see (1.139) of [45] for example). Since the sectional curvature of (M,g) ...

APpROXIMATE BUSEMANN-FELLER THEOREM 485 is a diffeomorphism, where Ns,.-1-l ~{av (x): x E 1-l and expx (sv (x)) c M+ for 0 :S ...

486 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION We have (I.49) Claim. If "(EN/ (H), where 0 < c::::; l (A, K,), then (I. ...

EQUIVALENCE CLASSES OF RAYS AND POINTS AT INFINITY 487 (note that [rs, rt]= 0 and 'Vrsrs = 0), equation (I.52) implies a2 1rtl ...

488 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION " ... Gromov has defined, in his lectures [3], the Tits' metric on the poin ...

EQUIVALENCE CLASSES OF RAYS AND POINTS AT INFINITY 489 We begin with an example on a compact manifold which contrasts to the f ...

490 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION In Lemma I.47, parts (a), (b), and (c) correspond to parts (ii), (iii), and ...

EQUIVALENCE CLASSES OF RAYS AND POINTS AT INFINITY 491 We have limLe = oo, e-+0 whereas the length along IR x {O} from (0, 0) ...

492 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION is distance nonincreasing (in particular, the map <Ps,t is continu- ous) ...

EQUIVALENCE CLASSES OF RAYS AND POINTS AT INFINITY 493 LEMMA I. 53. Let (Mn, g) be a complete noncom pact Riemannian mani- fol ...

494 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION By (i) we have . ds(p s) ( ai n s (p, s) '/3i n s (p, s)) hm ' =0; s--+oo S ...

NOTES AND COMMENTARY 495 Assuming Presumed Theorem I.50 is true, the definition of M ( oo) in (I.10) is equivalent to the defi ...

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APPENDIX J Solutions to Selected Exercises It's not what you know; it's what you can prove. From the movie "Training Day". SOL ...

498 J. SOLUTIONS TO SELECTED EXERCISES Hence, by (J.2) and (J.4), for all 1::::; k::::; m + 1 the distance of Xk to xo at time t ...

J. SOLUTIONS TO SELECTED EXERCISES 499 T/i in (25.30) now satisfies 0:::; TJ~ :::; ('Y!~)r 6 , and i/Ji in (25.31) now satisfies ...

500 J. SOLUTIONS TO SELECTED EXERCISES where r, (), z are cylindrical coordinates. That is, X is obtained by revolving Y about t ...