Volume 13, issue 2 (2009)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Equivariant Ricci flow with surgery and applications to finite group actions on geometric $3$–manifolds

Jonathan Dinkelbach and Bernhard Leeb

Geometry & Topology 13 (2009) 1129–1173
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].

Keywords
group action, Ricci flow, geometric manifold
Mathematical Subject Classification 2000
Primary: 57M60, 57M50
Secondary: 53C21, 53C44
References
Publication
Received: 7 July 2008
Revised: 9 January 2009
Accepted: 28 November 2008
Published: 27 January 2009
Proposed: Dave Gabai
Seconded: Colin Rourke, Tobias Colding
Authors
Jonathan Dinkelbach
Mathematisches Institut der LMU
Theresienstr. 39
80333 München
Germany
Bernhard Leeb
Mathematisches Institut der LMU
Theresienstr. 39
80333 München
Germany