1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

THE MAIN RESULT ON NONSINGULAR SOLUTIONS 201 (12) A rotationally symmetric n eckpinch (see Angenent and one of the authors [13 ...

202 32. NONSINGULAR SOLUTIONS ON CLOSED 3 -MANIFOLDS (B3) and (B4) (Sequential convergence to a flat or hyperbolic manifold) The ...

THE THREE CASES OF NONSINGULAR SOLUTIONS 203 Now, the closure C^3 of the complement of the union of all the immortal almost hy ...

204 32. NONSINGULAR SOLUTIONS ON CLOSED 3-MANIFOLDS 3.2. Compactness theorem for the NRF. The first main tool we shall use in th ...

THE THREE CASES OF NONSINGULAR SOLUTIONS 205 Each higher time derivative of r (t) is an integral of a polynomial expression of ...

206 32. NONSINGULAR SOLUTIONS ON CLOSED 3 -MANIFOLDS 3.4. Division of the study of nonsingular solutions into three cases. We sh ...

4. THE POSITIVE AND ZERO CASES OF NONSINGULAR SOLUTIONS 207 STEP 2. M is diffeomorphic to a space form. Since g 00 (t) has finit ...

208 32. NONSINGULAR SOLUTIONS ON CLOSED 3-MANIFOLDS On the other hand, since M admits a metric with positive scalar curvature, b ...

THE POSITIVE AND ZERO CASES OF NONSINGULAR SOLUTIONS 209 Let ti ~ J~: 'ljJ ( T) dT, let Xi satisfy (32.6), and let (M:;,,, g 0 ...

210 32. NONSINGULAR SOLUTIONS ON CLOSED 3-MANIFOLDS 5. The negative case-sequential limits must be hyperbolic In this section, b ...

NOTES AND COMMENTARY Next we compute that JM IR - rl dμ ~JM ((R-Rmin) + (r - Rmin)) dμ = 2A (r - Rmin), where A= Vol (g(t)). T ...

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CHAPTER 33 N oncompact Hyperbolic Limits And when your run is over jus t admit when it's at its end. From "'Till I Collapse" by ...

214 33. NONCOMPACT HYPERBOLIC LIMITS Main results on hyperbolic pieces In this section, after recalling some basic facts about ...

MAIN RESULTS ON HYPERBOLIC PIECES 215 manifold ( H^3 , h, x 00 ). By the definition of convergence and as a consequence of the ...

216 33. NONCOMPACT HYPERBOLIC LIMITS DEFINITION 33.2 (Stable hyperbolic limit). Let (M^3 , g(t)) be a nonsingular solution satis ...

MAIN RESULTS ON HYPERBOLIC PIECES 217 (2) For each t E [T, oo), llF (t)* g (t) - hl1iA llCk(1iA,h) ~ ~· Here, the immortal alm ...

218 33. NONCOMPACT HYPERBOLIC LIMITS In particular, int(MA (t)) is diffeomorphic to 7-l and for any k E N there exists tA,k E [t ...

HARMONIC MAPS PARAMETRIZING ALMOST HYPERBOLIC PIECES 219 1.4. Results on CMC surfaces and harmonic maps. When we have n-dimens ...

220 33. NONCOMPACT HYPERBOLIC LIMITS PROPOSITION 33.13 (Existence of harmonic parametrizations of almost hy- perbolic pieces). L ...