1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS 381 Applying the relative volume comparison theorem, which says -----,--VolB(x ...

382 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS Once we have (26.145), we may apply the maximum principle of Karp and Li ...

DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS 383 we obtain fl f e-ar2(x)77 (x) [\7f12 (x, t) dμ (x) dt lo 1B(0,2R) 1 11 ear ...

384 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS in B ( 0, p/VK) x (0, l]. By the gradient estimate (26.142), we have Ii ...

DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS 385 In particular, we have llf (·, 1)11 0 2,a S C13 in fJ (0, po/2) with respe ...

386 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS 5. Notes and commentary §1. The existence and positivity of the heat ker ...

Appendix G. Elementary Aspects of Metric Geometry I'm still at it, After-mathematics. From "Still D.R.E." by Dr. Dre featuring ...

388 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY books on metric and related Riemannian geometry are Gromov's [78] and Burago, Burag ...

METRIC SPACES AND LENGTH SPACES 389 REMARK G.1 (Scaling metric spaces). If (X, d) is a metric space, then so is (X, ad), where ...

390 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY If (X, dx) and (Y, dy) are metric spaces, then the product metric on X x Y is defin ...

METRIC SPACES AND LENGTH SPACES 391 (see Exercise G.6 below). If there are no paths in A between x and y, then the distance is ...

392 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY 1.1.3. Relations between metric spaces and length spaces. Given a (quasi-)metric sp ...

METRIC SPACES AND LENGTH SPACES 393 given by L(1) ~L 9 (1) ~ 1bl~= (u), 9 du. Let A denote the space of piecewise C^1 paths in ...

394 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY (Y, dy) as the infimum, over all metric spaces ( Z, dz) and isometric em- beddings ...

METRIC SPACES AND LENGTH SPACES (2) If f: (X, dx) --+ (Y, dy) is an E-isometry, then daH ((X, dx), (Y, dy )) < 2c. 395 REMA ...

396 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY From (G.8) we have By (yo,c:-^1 +c:) C N2ro (f (Bx (xo,c:-^1 ))); hence by (G.9), f ...

METRIC SPACES AND LENGTH SPACES 397 Then 9J1 is precompact in the collection of metric spaces with respect to the Gromov-Hausd ...

398 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY (TpX, dTpX) exist. Does there exist a (natural) embedding i : TpS ---+ TpX such tha ...

METRIC SPACES AND LENGTH SPACES 399 it in [18]). However, in general, for simply-connected complete noncom- pact manifolds wit ...

400 G. ELEMENTARY ASPECTS OF METRIC GEOMETRY REMARK G.31. (1) Note that the distance between two points x and y in the sphere sn ...