Geometrization of Three-Dimensional Orbifolds via Ricci Flow

Abstract

A three-dimensional closed orientable orbifold (with no bad suborbifolds) is known to have a geometric decomposition from work of Perelman along with earlier work of Boileau-Leeb-Porti and Cooper-Hodgson-Kerckhoff. We give a new, logically independent, unified proof of the geometrization of orbifolds, using Ricci flow. Along the way we develop some tools for the geometry of orbifolds that may be of independent interest.

20 figures, final version

References in corpus (4)

• Ricci flow with surgery on three-manifolds
• Roots in 3-manifold topology
• Locally Collapsed 3-Manifolds
• On the topology of locally volume collapsed Riemannian 3-orbifolds

Cited by in corpus (14)

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• Torus orbifolds, slice-maximal torus actions and rational ellipticity
• Isometric actions on spheres with an orbifold quotient
• Ricci flow from spaces with isolated conical singularities
• Non-negatively curved GKM orbifolds
• Positively curved Killing foliations via deformations
• Riemannian foliations on contractible manifolds
• The topology of the space of J-holomorphic maps to CP^2
• Four-orbifolds with positive isotropic curvature
• Compact manifolds of dimension $n\ge 12$ with positive isotropic curvature
• Comparison geometry of holomorphic bisectional curvature for Kaehler manifolds and limit spaces
• Localized Index and $L^2$-Lefschetz fixed point formula for orbifolds
• Regularized mean curvature flow for invariant hypersurfaces in a Hilbert space and its application to gauge theory
• A Floer homology invariant for $3$-orbifolds via bordered Floer theory