Geometrisierung in Dimension 3 und Geometrie singulärer Räume
Final Report Abstract
The context of our project is the Geometrization Program in dimension three. The Geometrization Conjecture, formulated and provided with substantial evidence by W. Thurston in the late 70's, asserts that every closed 3-manifold has a canonical topological decomposition into geometric pieces. It profoundly changed the landscape in 3-dimensional topology and established a close connection to differential geometry, especially to hyperbolic geometry and the theory of Kleinian groups. Shortly afterwards, in his important paper on 3-manifolds with positive Ricci curvature, R. Hamilton introduced a "heat flow" on Riemannian metrics, the so-called Ricci flow, and subsequently developed it as a possible approach to geometrization via geometric analysis. The breakthrough in realizing Hamilton's program was achieved by G. Perelman in 2003 and led him to a complete proof of the Geometrization Conjecture for 3-manifolds. With regard to finite smooth group actions on 3-manifolds, Thurston had formulated in 1981 an analogous conjecture for the geometrization of 3-dimensional orbifolds, the so-called "Orbifold Theorem" and outlined a strategy of proof. The two main themes of our project were Thurston's approach to the Orbifold Theorem and related questions about cone manifold structures and their deformations, and the approach using the Ricci flow with surgery opened up by Perelman. A detailed proof of the Orbifold Theorem in the general orientable case following Thurston's strategy has been given by M. Boileau, B. Leeb and J. Porti. A basic role is played by certain geometric structures with very special singularities, so-called cone manifold structures, which are used to interpolate between orbifold and smooth geometric structures. An important part of our project was devoted to the study of their deformation spaces. The local shape of the deformation space of hyperbolic cone manifold structures was determined by H. Weiß in case the cone angles are ≤ π. A global rigidity theorem for hyperbolic cone manifold structures was obtained by H. Weiß under the same assumption. Both results also apply in the spherical case. In another work the global shape of the space of cone manifold structures on a euclidean cone manifold with cone angles ≤ π was determined together with J. Porti. The deformation theory of hyperbolic cone manifolds with cone angles < 2π was settled in case the topological type is fixed. This result was extended together with G. Montcouquiol in a setting where a controlled change of the topological type is allowed. The spectral geometry of hyperbolic 3-manifolds with Dehn surgery type singularities was studied in cooperation with F. Pfäffle. Regarding the program of geometrizing 3-orbifolds via Ricci flow, J. Dinkelbach and B. Leeb gave a uniform Ricci flow proof for the fact that smooth finite group actions on spherical and hyperbolic 3-manifolds, and also on S2 x S1 are standard. Together with corresponding older results of Meeks and Scott for the other five Thurston geometries this implies that closed 3-orbifolds finitely covered by geometric ones are themselves geometric, settling a question posed by Thurston in 1982. D. Faessler and B. Leeb recently succeeded in carrying out some parts of the program in the general (not necessarily orientable) case. The most important result in this direction concerns the final geometric part of the geometrization argument, namely that locally collapsed Riemannian 3-orbifolds without bad 2-suborbifolds satisfy geometrization. Such collapsed orbifolds occur as the thin parts in the Ricci flow output at large times.
Publications
- The deformation theory of hyperbolic cone-3-manifolds with cone-angles less than 27T. preprint
H. Weiß
- Uniformization of small 3-orbifolds. C. R. Acad. Sci., Paris, Ser. I, Math. 332, No.1, 57-62 (2001)
M. Boileau, B. Leeb, J. Porti
- Local rigidity of 3-dimensional cone manifolds. Ph.D. thesis, Tübingen, 2002
H. Weiß
- Geometrization of S-dimensional orbifolds. Ann. Math. (2) 162, No. 1, 195-290 (2005)
M. Boileau, B. Leeb, J. Porti
- Local rigidity of 3-dimensional cone-manifolds. J. Differ. Geom. 71, No. 3, 437-506 (2005)
H. Weiß
- Deforming Euclidean cone 3-manifolds. Geom. Topol. 11, 1507-1538 (2007)
J. Porti, H. Weiß
- Global rigidity of 3-dimensional cone-manifolds. J. Differ. Geom. 76, No. 3,495-523 (2007)
H. Weiß
- Equivariant Ricci flow witfi surgery. Ph.D. thesis, München, 2008
J. Dinkelbach
- Geometrization of 3-dimensional manifold and Ricci flow: On Perelman's proof of the conjectures of Poincare and Thurston. Boll. Unione Mat. Ital. (9) 1, No. 1, 41-55 (2008)
B. Leeb
- Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds. Geom. Topol. 13, No. 2, 1129-1173 (2009)
J. Dinkelbach, B. Leeb
- The Laplacian on hyperbolic 3-manifolds with Dehn surgery type singularities. Commun. Anal. Geom. 17, No. 3, 505-528 (2009)
F. Pfäffle, H. Weiß
- The Teichmüller space of the punctured sphere and the deformation theory of hyperbolic 3-cone-manifolds. preprint, 2010
G. Montcouquiol, H. Weiß
