Geometry & Topology

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

Jonathan Dinkelbach and Bernhard Leeb

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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.)

group action Ricci flow geometric manifold


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.

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