Comparison Geometry

Mathematical Sciences Research Institute

doi:10.1017/9781009701891

This book documents the focus on a branch of Riemannian geometry called Comparison Geometry. The simple idea of comparing the geometry of an arbitrary Riemannian manifold with the geometries of constant curvature spaces has seen a tremendous evolution of late. This volume is an up-to-date reflection of the recent development regarding spaces with lower (or two-sided) curvature bounds. The content of the volume reflects some of the most exciting activities in comparison geometry during the year and especially of the Mathematical Sciences Research Institute's workshop devoted to the subject. Both survey and research articles are featured. Complete proofs are often provided, and in one case a new unified strategy is presented and new proofs are offered. This volume will be a valuable source for advanced researchers and those who wish to learn about and contribute to this beautiful subject.

Review of the hardback:‘… a beautiful, comprehensive and up-to-date collection of expository experts in the field.’

Source: European Mathematical Society

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Comparison Geometry

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Contents

Frontmatter
pp i-vi

Contents
pp vii-viii

Preface
pp ix-x

Injectivity Radius Estimates and Sphere Theorems
pp 1-48

Scalar Curvature and Geometrization Conjectures for 3-Manifolds
pp 49-82

Aspects of Ricci Curvature
pp 83-98

A Genealogy of Noncompact Manifolds of Nonnegative Curvature: History and Logic
pp 99-134

Differential Geometric Aspects of Alexandrov Spaces
pp 135-148

Collapsing with No Proper Extremal Subsets
pp 149-156

Construction of Manifolds of Positive Ricci Curvature with Big Volume and Large Betti Numbers
pp 157-164

A Complete Riemannian Manifold of Positive Ricci Curvature with Euclidean Volume Growth and Nonunique Asymptotic Cone
pp 165-166

Convergence Theorems in Riemannian Geometry
pp 167-202

Applications of Quasigeodesics and Gradient Curves
pp 203-220

The Comparison Geometry of Ricci Curvature
pp 221-262

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