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

Dinkelbach, Jonathan¹ ; Leeb, Bernhard¹

¹ Mathematisches Institut der LMU, Theresienstr. 39, 80333 München, Germany

Geometry & topology, Tome 13 (2009) no. 2, pp. 1129-1173.

Résumé

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].

Détail

DOI : 10.2140/gt.2009.13.1129

Keywords: group action, Ricci flow, geometric manifold

Bibliographie

[1] R Bamler, Ricci flow with surgery, PhD thesis, München (2007)

[2] M Boileau, B Leeb, J Porti, Geometrization of $3$–dimensional orbifolds, Ann. of Math. $(2)$ 162 (2005) 195

[3] E J Brody, The topological classification of the lens spaces, Ann. of Math. $(2)$ 71 (1960) 163

[4] J Cheeger, D Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. $(2)$ 96 (1972) 413

[5] T H Colding, W P Minicozzi Ii, Estimates for the extinction time for the Ricci flow on certain $3$–manifolds and a question of Perelman, J. Amer. Math. Soc. 18 (2005) 561

[6] T H Colding, W P Minicozzi Ii, Width and finite extinction time of Ricci flow, Geom. Topol. 12 (2008) 2537

[7] J Dinkelbach, Equivariant Ricci flow with surgery, PhD thesis, München (2008)

[8] D B A Epstein, Curves on $2$–manifolds and isotopies, Acta Math. 115 (1966) 83

[9] D Gromoll, W Meyer, On complete open manifolds of positive curvature, Ann. of Math. $(2)$ 90 (1969) 75

[10] K Grove, H Karcher, How to conjugate $C^{1}$–close group actions, Math. Z. 132 (1973) 11

[11] K Grove, K Shiohama, A generalized sphere theorem, Ann. Math. $(2)$ 106 (1977) 201

[12] R S Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255

[13] A Hatcher, Notes on basic $3$–manifold topology (2000)

[14] M W Hirsch, S Smale, On involutions of the $3$–sphere, Amer. J. Math. 81 (1959) 893

[15] B Kleiner, J Lott, Notes on Perelman's papers, Geom. Topol. 12 (2008) 2587

[16] G R Livesay, Involutions with two fixed points on the three-sphere, Ann. of Math. $(2)$ 78 (1963) 582

[17] W H Meeks Iii, P Scott, Finite group actions on $3$–manifolds, Invent. Math. 86 (1986) 287

[18] W H Meeks Iii, S T Yau, Group actions on $\mathbf{R}^{3}$, from: "The Smith conjecture (New York, 1979)", Pure Appl. Math. 112, Academic Press (1984) 167

[19] W Meyer, Toponogov's theorem and applications, Lecture notes (1989)

[20] J Morgan, G Tian, Completion of the proof of the Geometrization Conjecture

[21] J Morgan, G Tian, Ricci flow and the Poincaré conjecture, Clay Math. Monogr. 3, Amer. Math. Soc. (2007)

[22] G D Mostow, Quasi-conformal mappings in $n$–space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. (1968) 53

[23] J Munkres, Differentiable isotopies on the $2$–sphere, Michigan Math. J. 7 (1960) 193

[24] R S Palais, Equivalence of nearby differentiable actions of a compact group, Bull. Amer. Math. Soc. 67 (1961) 362

[25] G Perelman, The entropy formula for the Ricci flow and its geometric applications

[26] G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

[27] G Perelman, Ricci flow with surgery on three-manifolds

[28] G Prasad, Strong rigidity of $\mathbf{Q}$–rank $1$ lattices, Invent. Math. 21 (1973) 255

[29] P Scott, The geometries of $3$–manifolds, Bull. London Math. Soc. 15 (1983) 401

[30] J P Serre, Linear representations of finite groups, Graduate Texts in Math. 42, Springer (1977)

[31] W P Thurston, Three-manifolds with symmetry, Preprint

[32] W P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. $($N.S.$)$ 6 (1982) 357

[33] W P Thurston, Three-dimensional geometry and topology. Vol. 1 (editor S Levy), Princeton Math. Ser. 35, Princeton Univ. Press (1997)