1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

TOPOLOGY AND GEOMETRY OF HYPERBOLIC 3-MANIFOLDS isi 7ri (E^2 ) Iker (i*) must be isomorphic to either 0, z, or z x Z.^2 By the ...

182 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS LEMMA 31.21 (Separating surfaces and incompressibility). Suppose that E^2 is a close ...

2. TOPOLOGY AND GEOMETRY OF HYPERBOLIC 3-MANIFOLDS 183 PROOF. Let i : Vr -+ 7-l denote the inclusion map. If the lemma is false, ...

184 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS We claim that case (Al) cannot happen. Suppose ¢ 1 and </>2 are two hyperbolic ...

THE MARGULIS LEMMA AND HYPERBOLIC CUSPS 185 In case (B2), the torus T lies in B. By the Jordan- Brouwer separation theorem, we ...

186 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS is a bijection. If JC is an end-complementary compact set, then we call each element ...

3. THE MARGULIS LEMMA AND HYPERBOLIC CUSPS 18 7 some of the horoballs are tangent to each other. In a maximal cusp end, for each ...

188 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS Now we turn to study the €-thin and €-thick parts of a hyperbolic manifold. DEFINITI ...

THE MARGULIS LEMMA AND HYPERBOLIC CUSPS 189 set 1-l[e,oo) is not always end-complementary. This happens when there is a compac ...

190 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS PROOF. Let (Hn, h) b e a complete hyperbolic manifold. Since its fundamental group i ...

THE MARGULIS LEMMA AND HYPERBOLIC CUSPS 191 THEOREM 31.46 (Volume is bounded below by the number of cusp ends). If (1i^3 , h) ...

192 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS Mostow rigidity In this section we state a special case of t he Mostow rigidity th ...

SEIFERT FIBERED MANIFOLDS AND GRAPH MANIFOLDS 193 Seifert fibered manifolds and graph manifolds In this section we give some to ...

194 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS Graph manifolds are generalizations of Seifert fibered manifolds and can be defined ...

NOTES AND COMMENTARY 195 For a proof of the Jordan- Brouwer separation theorem, seep. 89 of Guillemin and Pollack [131]. Using ...

196 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS §4. For the proof of the Mostow ri gidity theorem, see Mostow [258] and Prasad [332] ...

CHAPTER 32 Nonsingular Solutions on Closed 3-Manifolds Maybe this time is forever ... From "Father Figure" by George Michael B ...

198 32. NONSINGULAR SOLUTIONS ON CLOSED 3 -MANIFOLDS 1.1. The flip side-singular solutions and singularity models. Nonsingular s ...

l. INTRODUCTION 199 By the work of Angenent and one of the authors [13], one may have for example a "rotationally symmetric" ~P^ ...

200 32. NONSINGULAR SOLUTIONS ON CLOSED 3-MANIFOLDS The next chapter i s devoted to the proof that t he inclusion maps of such p ...