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El. knyga: Ricci Flow and Geometrization of 3-Manifolds

John Morgan, Translated by Frederick Fong

Formatas: 150 pages
Serija: University Lecture Series
Išleidimo metai: 03-Sep-2010
Leidėjas: American Mathematical Society
ISBN-13: 9781470416485

Kitos knygos pagal šią temą:

Mathematics

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Formatas: 150 pages
Serija: University Lecture Series
Išleidimo metai: 03-Sep-2010
Leidėjas: American Mathematical Society
ISBN-13: 9781470416485

Kitos knygos pagal šią temą:

Mathematics

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This book is based on lectures given at Stanford University in 2009. The purpose of the lectures and of the book is to give an introductory overview of how to use Ricci flow and Ricci flow with surgery to establish the Poincare Conjecture and the more general Geometrization Conjecture for 3-dimensional manifolds. Most of the material is geometric and analytic in nature; a crucial ingredient is understanding singularity development for 3-dimensional Ricci flows and for 3-dimensional Ricci flows with surgery. This understanding is crucial for extending Ricci flows with surgery so that they are defined for all positive time. Once this result is in place, one must study the nature of the time-slices as the time goes to infinity in order to deduce the topological consequences. The goal of the authors is to present the major geometric and analytic results and themes of the subject without weighing down the presentation with too many details. This book can be read as an introduction to more complete treatments of the same material. (ULECT/53)

Preface

Part 1 Overview

(26)

Lecture 1

(1)

Geometric manifolds

(1)

Thurston manifolds

(1)

The theorems

(2)

Lecture 2

(1)

Basics of Riemannian geometry

(1)

Basics of Ricci flow

(2)

Canonical Neighborhoods

(3)

Lecture 3

(1)

More on Canonical Neighborhoods

(1)

Surgery on Ricci flow

(2)

Topological effects of surgery

(1)

Lecture 4

(1)

More structure (geometric and analytic) of Canonical Neighborhoods

(1)

Finite-time extinction

(3)

Lecture 5

(1)

Geometric limits

(1)

Hyperbolic limits

(1)

The thin part

(1)

Alexandrov spaces

(4)

Summary of Part 1

(2)

Part 2 Non-collapsing Results for Ricci Flows

(30)

Lecture 6

(1)

Geometric limits in the context of Ricci flow

(2)

Sketch of proof of the convergence theorem

(2)

Lecture 7

(1)

Non-collapsing: the statement

(1)

The L-function and L-geodesics

(3)

Lecture 8

(1)

The L-exponential map

(1)

Jacobi fields and the differential of L-exp

(3)

Lecture 9

(1)

Harnack's inequality

(1)

Relation of H(X) to L-geodesics

(3)

Lecture 10

(1)

More derivative estimates for L

(2)

Hessian inequality

(2)

Lecture 11

(1)

Monotonicity

(1)

Example of Rn

(1)

Non-collapsing of reduced volume

(2)

Lecture 12

(1)

Non-collapsing

(1)

Completion of proof

(4)

Part 3 k-solutions

(32)

Lecture 13

(1)

Curvature pinching in dimension 3

(1)

Shrinking solitons

(4)

Lecture 14

(1)

Study of the length functions in a k-solution

(1)

Extensions of the inequalities

(1)

Convergence as τ→∞

(3)

Lecture 15

(1)

Proof of the existence of an asymptotic gradient shrinking soliton

(3)

Enhanced gradient shrinking solitons

(3)

Lecture 16

(1)

Toponogov's splitting theorem

(1)

Classification of asymptotic gradient shrinking solitons

(4)

Lecture 17

(1)

Asymptotic volume ratio and asymptotic curvature

(1)

Asymptotic curvature of a k-solution

(1)

Asymptotic volume ratio for a k-solution

(3)

Lecture 18

(1)

Compactness of the space of k-solutions

(1)

Proof of the compactness theorem for k-solutions

(3)

Lecture 19

(1)

Review of compactness of 3-dimensional k-solutions

(1)

Qualitative properties of k-solutions

(1)

Geometry of 3-dimensional k-solutions

(2)

Part 4 The Canonical Neighborhood Theorem

(16)

Lecture 20

(1)

Blow-up limits

(1)

Canonical neighborhood theorem

(5)

Lecture 21

(1)

Completion of the proof of the canonical neighborhood theorem

(1)

Step 2 of proof

(1)

Step 3 of proof

(4)

Lecture 22

101

(1)

Review of proof

101

(1)

Additive distance inequality

101

(4)

Part 5 Ricci Flow with Surgery

105

(20)

Lecture 23

107

(1)

What happens at Tmax?

107

(1)

ε-horns

108

(1)

Structure of Ω

108

(1)

Topological description of surgery

109

(2)

Lecture 24

111

(1)

Geometric surgery on a Ricci flow

111

(1)

Surgery (refined)

112

(1)

The standard solution

112

(3)

Lecture 25

115

(1)

Existence of Ricci flow with surgery defined for all time: the statement and outline of proof

115

(2)

Noncollapsing

117

(4)

Lecture 26

121

(1)

ε-canonical neighborhood threshold parameter

121

(2)

Discreteness of the surgery times

123

(2)

Part 6 Behavior as t→∞

125

(24)

Lecture 27

127

(1)

Recap of results of previous parts

127

(1)

Normalized volume and scalar curvature at infinity

127

(4)

Lecture 28

131

(1)

Hyperbolic limits

131

(1)

Analytic results for large time

132

(3)

Lecture 29

135

(1)

Permanence of the hyperbolic pieces

135

(1)

Hyperbolic towers

136

(3)

Lecture 30

139

(1)

Incompressibility of the boundary tori

139

(1)

Structure of Mt, thin(w)

139

(2)

Lecture 31

141

(1)

The relative version of the Geometrization Conjecture

141

(1)

Proof that the theorem implies Geometrization Conjecture

142

(1)

Study of Mt, thin(w)

142

(3)

Lecture 32

145

(1)

The structure of sufficiently volume collapsed 3-manifolds

145

(1)

Gromov-Hausdorff limits

145

(1)

Alexandrov spaces

145

(2)

Structure of the ρ-1 (Xn) B Xn, ρ(Xn)) when the limit has dimension 1

147

(1)

Structure of the ρ-1 (Xn) B (Xn, ρ(Xn)) when the limit has dimension 2

147

(1)

The global structure

148

(1)

Bibliography

149

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