1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

SHARAFUTDINOV RETRACTION THEOREM 461 1.2.· Manifolds with nonnegative sectional curvature. When complete noncompact manifolds ...

462 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION PROPOSITION I.16 (Sublevel set of bp is compact and totally convex). Let (M ...

SHARAFUTDINOV RETRACTION THEOREM 463 DEFINITION l.21 (Soul). A submanifold S of a complete noncompact manifold (Mn, g) is call ...

464 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION PROOF. Define 1 = -bp - a2 : M --+ JR and C = (-bp)-^1 ([a2, oo )) = 1-^1 ( ...

THE EXISTENCE OF ASYMPTOTIC CONES 465 THEOREM I.25. Let (Mn,g) be a complete noncompact Riemannian manifold with nonnegative s ...

466 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION by t, taking the limit as t---+ oo, and applying part (i), we obtain ( a^2 ...

THE EXISTENCE OF ASYMPTOTIC CONES^467 Now we give a PROOF OF THEOREM 1.26. We claim that (M, A.^2 g) converges to the Euclidea ...

468 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION when>.. is sufficiently small. This and (I.11) imply dGH (N;..,B;..2g (O ...

MONOTONICITY OF SPHERES WITHIN THE INJECTIVITY RADIUS 4(39 endowed with the subspace metric, which we denote by ds(p,s)· The m ...

470 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION Then/ (0) = ao (t) = 'Ps,t (x) and/ (p) = ap (t) = 'Ps,t (y). Clearly L (1) ...

MONOTONICITY OF SPHERES WITHIN THE INJECTIVITY RADIUS 471 so that (I.21) dv di UI^2 (ao ( v)) ::=:: r (ao^2 (v)) IUI^2 (ao (v) ...

472 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION 4. Critical point theory and properties of distance spheres Continuing with ...

CRITICAL POINT THEORY AND PROPERTIES OF DISTANCE SPHERES 473 The set V' *r C SM is locally compact. In particular, if M is com ...

474 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION EXAMPLE I.32 (Spheres and tori). (i) The sphere Sn (p) of radius p. Given p ...

CRITICAL POINT THEORY AND PROPERTIES OF DISTANCE SPHERES 475 PROOF. (1) If case. Suppose that VE s;i-^1 , c > 0, and c > ...

476 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION for all so E (0, c:) such that (r o a)' (so) exists. Therefore ( r o a) ( s ...

CRITICAL POINT THEORY AND PROPERTIES OF DISTANCE SPHERES 477 for all i E N. Then for all i there exist Vi, Wi E (\7 * r) (Xi) ...

478 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION and (I.35) We shall prove a uniform positive lower bound for ¢i· In this re ...

CRITICAL POINT THEORY AND PROPERTIES OF DISTANCE SPHERES 479 EXERCISE I.38. Under the hypotheses in the proof of Lemma I.37 ab ...

480 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION Our interest in Lipschitz hypersurfaces stems from the following. ·For the ...