1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

£-JACOBI FIELDS AND THE £-EXPONENTIAL MAP 357 the proof of the Riemannian index lemma. We will give the details of the proof e ...

358 7. THE REDUCED DISTANCE PROOF. Suppose limr--->O ft~~ (0) = V; the hypothesis implies by Lemma 7.92 that there is some E ...

£.-JACOBI FIELDS AND THE £-EXPONENTIAL MAP 359 Rearranging terms and integrating by parts, we express this as follows. De- fin ...

360 7. THE REDUCED DISTANCE where the exponential map is restricted to inside the cut locus in the tangent space. Then the volum ...

£-JACOBI FIELDS AND THE £-EXPONENTIAL MAP 361 To get a feeling of the £-Jacobian, we calculate an example. EXAMPLE 7.103 (£-Ja ...

362 7. THE REDUCED DISTANCE PROPOSITION 7.105 (Time-derivative of £-Jacobian). Let (Mn, g (r)), r E [O, T] , be a solution of th ...

WEAK SOLUTION FORMULATION 363 Now we compute that the evolution of the £-Jacobian along /'Vis given by = ~ d~'T=T logdet((Ei,E ...

364 7. THE REDUCED DISTANCE Lipschitz functions and other aspects of real analysis on IRn· is the book by Evans and Gariepy [139 ...

WEAK SOLUTION FORMULATION 365 LEMMA 7.112 (Hausdorff dimension of a Lipschitz graph). Let (.Mr, § 1 ) and (.M~i, §2) be two Ri ...

366 7. THE REDUCED DISTANCE Let U 1 be an open neighborhood of the support of v satisfying U1 CC U.^16 We can apply Theorem 1 in ...

WEAK SOLUTION FORMULATION 367 (the first derivative) such that If (y) - f ( x) - D f ( x). (y - x) - ~ (y - x) T. D^2 f ( x). ...

368 7. THE REDUCED DISTANCE where a > 0 is chosen so that JB(l) rJdx = 1. This function is C^00 with support contained in B ( ...

WEAK SOLUTION FORMULATION 369 Since Dl (0) = 0, given any El > 0 there exists o > 0 such that ll (y)I :::; Ei lvl whenev ...

370 7. THE REDUCED DISTANCE (iii) For 0::::; .\::::; 1 and x,y EB (r - c), we compute fi; (.Ax+ (1 - .A) y) = 2- r f (.A (x + z) ...

WEAK SOLUTION FORMULATION 371 then there exists a C^2 function 'I/; (x) defined on Bp (2r) c TpM such that (f o exp_;^1 +'I/;) ...

372 7. THE REDUCED DISTANCE which contradicts (7.143). This completes the proof of the claim and part (ii). (iii) By hypothesis, ...

WEAK SOLUTION FORMULATION 373 (ii) Let q E B (p, r) and let coordinates x be the same as in (i), where \J\Jf (q) = D^2 f (0) e ...

374 7. THE REDUCED DISTANCE (iii) A continuous function f : M ---+ R is said to satisfy D.f :::; k in the viscosity sense for so ...

WEAK SOLUTION FORMULATION 375 D^2 f ( x) uniformly on Bp ( !r) \ W 01. This in turn implies that b..9 fc:k ___,. b..9 f unifor ...

376 7. THE REDUCED DISTANCE 9.5.1. Weak differentiability of the distance function d on a Riemannian manifold. Let (.Mn, g) be a ...