Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery.

*(English)*Zbl 0919.57009The 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 WeitzenbĂ¶ck 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.

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 WeitzenbĂ¶ck 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.

Reviewer: Ser Peow Tan (Singapore)