1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

ESTIMATES FOR CHANGING DISTANCES 41 denote the lim inf of forward difference quotients. We have a similar defini- tion for^8 a ...

42 18. GEOMETRIC TOOLS AND POINT PICKING METHODS to 0, we have a-1 at t=to dg(t)(x,xo) 2: dt di t=to Lg(t) (rJoo) 2: min ( ~ 1 ~ ...

ESTIMATES FOR CHANGING DISTANCES 43 where, as above, Z (t) is the compact set of all unit speed minimal geodesics joining xo t ...

44 18. GEOMETRIC TOOLS AND POINT PICKING METHODS (3) f'v (so) = expx (V), and ( 4) gr / r=O Irv ( S) = V ( s) for V E B ( c). Fo ...

We compute b.gf(x) n-l ESTIMATES FOR CHANGING DISTANCES = L V'V' f (Ei, Ei) + V'V' f (1^1 (so),1' (so)) i=l = n-l tt 8r2 EJ2 I ...

46 (18.9) (18.10) GEOMETRIC TOOLS AND POINT PICKING METHODS (a) If dg(to) (xo, x1) 2: 2ro, then the time derivative of the dis ...

ESTIMATES FOR CHANGING DISTANCES 47 Combining this with (18.12), we obtain (! -~g(t)) dg(t)(x,xo)/t=to 2: -(n-1) (~Kro + : 0 ) ...

48 18. GEOMETRIC TOOLS AND POINT PICKING METHODS (2)(b) When dg(to) (xo, x1) < 2ro, the minimal geodesic')' (s) in (18.12) jo ...

SPATIAL POINT PICKING METHODS 49 since f 13 Re ({3' ( s) , {3' ( s)) ds :::; ( n - 1) KL (/3). REMARK 18.9. The above results, ...

50 18. GEOMETRIC TOOLS AND POINT PICKING METHODS then there exist a sequence of points {xi}: 1 in M with d (xi, 0) --too and seq ...

SPATIAL POINT PICKING METHODS 51 (3) d (xi, 0) -t oo, (4) the balls B (xi, ri) are disjoint. PROOF. (1) Since supM R = oo, the ...

52 18. GEOMETRIC TOOLS AND POINT PICKING METHODS Define fi ~ d(xi,8B(yi,1)) and take Ei E (0,1) to be chosen below. For any x EB ...

SPATIAL POINT PICKING METHODS 53 REMARK 18.16. Thus, relative to the curvature at the center, we have uniformly bounded curvat ...

54 18. GEOMETRIC TOOLS AND POINT PICKING METHODS 2.4.l. Point picking criteria for the existence of a line in the limit. The fol ...

SPATIAL POINT PICKING METHODS 55 REMARK 18.20. In particular, the hypotheses of the theorem imply that ASCR (9 (0)) = oo. Note ...

56 18. GEOMETRIC TOOLS AND POINT PICKING METHODS monotonicity principle, i.e., Lemma G.38,^9 then implies that the Euclidean com ...

SPACE-TIME POINT PICKING WITH RESTRICTIONS 57 nonnegative sectional curvature and 0 EM, then there are at most a finite number ...

58 18. GEOMETRIC TOOLS AND POINT PICKING METHODS PROOF. We shall construct by recursion a finite sequence of points {(yi, ti)} w ...

SPACE-TIME POINT PICKING WITH RESTRICTIONS 59 Since A:::; Ai (CT) :::; 23 ~ 4 ( C + B(ti - to)-^1 ), by (18.37) we obtain 1 4 ...

60 18. GEOMETRIC TOOLS AND POINT PICKING METHODS Then j (18.42) R (y, t) :S 2R(y*, t*) PROOF. ·we argue by contradiction. Suppos ...