Geometry & Topology

Equivariant Ricci flow with surgery and applications to finite group actions on geometric $3$–manifolds

Jonathan Dinkelbach and Bernhard Leeb

Full-text: Open access

Abstract

We apply an equivariant version of Perelman’s Ricci flow with surgery to study smooth actions by finite groups on closed 3–manifolds. Our main result is that such actions on elliptic and hyperbolic 3–manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [Invent. Math. 86 (1986) 287-346], it follows that such actions on geometric 3–manifolds (in the sense of Thurston) are always geometric, ie there exist invariant locally homogeneous Riemannian metrics. This answers a question posed by Thurston [Bull. Amer. Math. Soc. (N.S.) 6 (1982) 357-381].

Article information

Source
Geom. Topol., Volume 13, Number 2 (2009), 1129-1173.

Dates
Received: 7 July 2008
Revised: 9 January 2009
Accepted: 28 November 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800226

Digital Object Identifier
doi:10.2140/gt.2009.13.1129

Mathematical Reviews number (MathSciNet)
MR2491658

Zentralblatt MATH identifier
1181.57023

Subjects
Primary: 57M60: Group actions in low dimensions 57M50: Geometric structures on low-dimensional manifolds
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

Keywords
group action Ricci flow geometric manifold

Citation

Dinkelbach, Jonathan; Leeb, Bernhard. Equivariant Ricci flow with surgery and applications to finite group actions on geometric $3$–manifolds. Geom. Topol. 13 (2009), no. 2, 1129--1173. doi:10.2140/gt.2009.13.1129. https://projecteuclid.org/euclid.gt/1513800226


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