1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

HARMONIC MAPS PARAMETRIZING ALMOST HYPERBOLIC PIECES 221 Since <Pi gi converges to hl1-l A/2 in C^00 , we have that i-+cx:: ...

222 33. NONCOMPACT HYPERBOLIC LIMITS PROPOSITION 33.I4 (Continuing almost isometric harmonic parametrizations of hyperbolic piec ...

HARMONIC MAPS PARAMETRIZING ALMOST HYPERBOLIC PIECES 223 Before proving Claim 1, we first prove the following: Statement A. Le ...

224 33. NONCOMPACT HYPERBOLIC LIM ITS The following convergence, which proves closedness, is essent ially a consequence of the p ...

HARMONIC MAPS PARAMETRIZING ALMOST HYPERBOLIC PIECES 225 for 0 :::; j :::; k. To see this, note that analogously to (33.15) we ...

226 33. NONCOMPACT HYPERBOLIC LIMITS (33.11). Then, by the fact that F 2 (t) (8HA) = F 1 (t) (8HA) from Statement A and by the u ...

PROOF OF THE STABILITY OF HYPERBOLIC LIMITS 227 By property (1), the sequence of pointed closed Riemannian 3-manifolds (33.18) ...

228 33. NONCOMPACT HYPERBOLIC LIMITS Now suppose that the proposition is false for some A E (0, A] and some k. Without loss of g ...

PROOF OF THE STABILITY OF HYPERBOLIC LIMITS 229 To see (33.26), first note that by (33.24) we have (33.28) for all i. Moreover ...

230 33. NONCOMPACT HYPERBOLIC LIMITS conclude that with respect to h and h the map w 00 is a harmonic embedding satisfying both ...

PROOF OF THE STABILITY OF HYPERBOLIC LIMITS 231 STEP 1. Reduction to a claim. Theorem 33.17 is a consequence of the following. ...

232 33. NONCOMPACT HYPERBOLIC LIMITS This simple fact follows directly from Corollary 31.47 since the image of such a 2-torus un ...

PROOF OF THE STABILITY OF HYPERBOLIC LIMITS 233 the fact that the volumes of p(k) (t(k)) (Hi;k) are uniformly bounded from b e ...

234 33. NONCOMPACT HYPERBOLIC LIMITS will be chosen to be sufficiently small. Regarding the complement M -Mo,Ao(t) (t) of the fi ...

PROOF OF THE STABILITY OF HYPERBOLIC LIMITS 235 is a harmonic diffeomorphism satisfying the CMC boundary conditions and that ( ...

236 33. NONCOMPACT HYPERBOLIC LIMITS the corresponding hyperbolic limits (H~, hb) E ..IJIJPb(M, g (t)), and the correspond- ing ...

INCOMPRESSIBILITY OF BOUNDARY TORI 237 (4) harmonic diffeomorphisms Fa (t) : (Ha)Aa(t) ---t Ma,Aa(t)(t) satisfying the CMC bou ...

238 33. NONCOMPACT HYPERBOLIC LIMITS the function A : [T 0 , oo) --+ (0, A) is strictly decreasing in the definition of stabilit ...

INCOMPRESSIBILITY OF BOUNDARY TORI 239 THEOREM 33.25 (Meeks and Yau). Let (N^3 , g) be a compact Riemannian 3- manifold with b ...

240 33. NONCOMPACT HYPERBOLIC LIMITS Fix any time t 0 > 0, which will be chosen sufficiently large. Since the boundary torus ...