1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

SPACE-TIME POINT PICKING WITH RESTRICTIONS 61 be the maximal time interval such that for any t E ( t$, t*], (18.46) dh(t) (w,p ...

62 18. GEOMETRIC TOOLS AND POINT PICKING METHODS We can now apply Theorem 18.7(2) to get that for any t E [t$, t#], :tdh(t) (w,p ...

NECKS IN MANIFOLDS WITH POSITIVE SECTIONAL CURVATURE 63 in the cle- 1 l+l_topology^12 to a piece of the cylinder metric 9sn-l ...

64 18. GEOMETRIC TOOLS AND POINT PICKING METHODS 4.2. The diameter of a level set of a Busemann function in an c:-neck. The foll ...

NECKS IN MANIFOLDS WITH POSITIVE SECTIONAL CURVATURE 65 Let A be a constant to be chosen later, and E ( n) will be chosen so t ...

66 18. GEOMETRIC TOOLS AND POINT PICKING METHODS close to that of the standard cylinder, we have that, provided A 2: 51f,^17 the ...

NECKS IN MANIFOLDS WITH POSITIVE SECTIONAL CURVATURE 67 If c: ( n) > 0 is small enough, since the geometry of the neck 91 i ...

68 18. GEOMETRIC TOOLS AND POINT PICKING METHODS 4.3. Bounding radii of farther c:-necks by radii of closer E-necks. We now esti ...

LOCALIZED NO LOCAL COLLAPSING THEOREM 69 5.1. Statement of the main theorem. Recall that, as compared to Definition 19.1 below ...

70 18. GEOMETRIC TOOLS AND POINT PICKING METHODS 5.2. Proof of Theorem 18.36. The proof is similar to the proof of the 'weakened ...

LOCALIZED NO LOCAL COLLAPSING THEOREM 71 (here we used 11,^1 1nr^2 < c1 (n) :::; !). From this and (18.62), we conclude tha ...

72 18. GEOMETRIC TOOLS AND POINT PICKING METHODS We can estimate f (q, 1) by the reduced length of the concatenated path 'Y1 U ' ...

LOCALIZED NO LOCAL COLLAPSING THEOREM such that f(u) ~I 1 if u E (-oo, !e^1 -n], 0::; ¢/(u)::; 12exp (en+l + n) if u E (!e^1 - ...

74 18. GEOMETRIC TOOLS AND POINT PICKING METHODS (3) There exists y E B 9 (~) (xo, e^1 -n) such that H(~)=h(y,~). (4) At any poi ...

LOCALIZED NO LOCAL COLLAPSING THEOREM Hence, for a minimal £-geodesic"( from (x,O) to (w,r), we have L(w,r)=2v'T for vs(Rg('Y( ...

76 18. GEOMETRIC TOOLS AND POINT PICKING METHODS IRc(z, t) I ::;; n-1 for all z E Bg(t) (xo, ~e^1 -n), we can apply Theorem 18.7 ...

NOTES AND COMMENTARY 77 HINT. Just choose Ki sufficiently large. Note that the manifolds need not even be complete. §4. In App ...

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CHAPTER 19 Chapter 19. Geometric Properties of /\;-Solutions Will the wind ever remember the names it has blown in the past? Fr ...

80 19. GEOMETRIC PROPERTIES OF K:-SOLUTIONS In §6 we also concentrate on dimension 3 and prove the 11;-gap theorem for 3-dimensi ...