# zbMATH — the first resource for mathematics

Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. (English) Zbl 0919.57009
The authors extend the rigidity theorem of Weil and Garland for complete, finite volume hyperbolic manifolds of dimension at least 3 to a class of finite volume, orientable 3-dimensional hyperbolic cone-manifolds, that is, hyperbolic structures on 3-manifolds with cone-like singularities along a knot or link. Specifically, they show that such structures are locally rigid if the cone angles are fixed and all cone angles are at most $$2\pi$$. Furthermore, they show that in this case, the deformation space of cone-manifold structures is locally parametrized by the set of cone angles. The main theorem proved by the authors is an $$L^2$$ rigidity theorem (Theorem 1.1 of the paper) as follows:
Theorem 1.1: Let $$\overline M$$ be a finite volume, 3-dimensional hyperbolic cone-manifold, whose singular locus $$\Sigma$$ is a knot or link. Let $$M$$ denote the open, incomplete hyperbolic manifold $$\overline M-\Sigma$$. If all the cone angles along $$\Sigma$$ are at most $$2\pi$$, then every closed $$L^2$$ form in $$\Omega^1 (M,E)$$ represents the trivial cohomology class in $$H^1(M;E)$$.
An infinitesimal deformation of a hyperbolic structure on the manifold $$M$$ can be represented by an $$E$$-valued 1-form where $$E$$ is the flat vector bundle of local Killing vector fields. The authors show that any infinitesimal deformation preserving cone angles can be represented by an $$L^2$$ form. It then follows from theorem 1.1 that there are no infinitesimal deformations of the hyperbolic structure on $$M$$ keeping the cone angles fixed. The proof of theorem 1.1 in the paper involves the use of Hodge theory and a Weitzenböck type formula. It is similar to the approach of Calabi, Weil and Matsushima-Murakami where it was shown that for a closed oriented manifold $$M$$ every class in $$H^1(M;E)$$ is represented by a unique harmonic form $$\omega$$ which is both closed and co-closed and with the boundary term for the expansion of $$(d\omega,d\omega) + (\delta\omega, \delta\omega)$$ equal to 0. In the case of cone-manifolds, the analysis is more subtle and the authors show that any cohomology class can be represented by a form in $$\Omega^1(M;E)$$ which is closed and co-closed and with controlled behaviour near the singular locus $$\Sigma$$; roughly, the boundary terms should approach zero as the distance of $$\Sigma$$ approaches zero. More precisely, they show that in each cohomology class there is a unique representative which is closed, co-closed and traceless, and differs from any standard form by the derivative of the $$L^2$$ section (theorem 2.7).
The authors also conjecture that hyperbolic Dehn surgery space is star-like with respect to the rays from the origin to infinity. This would imply that if a collection of cone angles is realizable, then all smaller cone angles are realizable and in particular, one would be able to deform the cone structure to a complete finite volume hyperbolic structure on the complement of the singular locus. This would prove a global rigidity theorem for hyperbolic cone-manifolds extending the global rigidity theorem of Mostow-Prasad.

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds 57R65 Surgery and handlebodies
##### Keywords:
$$L^2$$ rigidity theorem; hyperbolic structure
Full Text: