Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery

Abstract

The local rigidity theorem of WeiI [28] and Garland [12] for complete, finite volume hyperbolic manifolds states that there is no non-trivial deformation of such a structure through complete hyperbolic structures if the manifold has dimension at least 3. If the manifold is closed, the condition that the structures be complete is automatically satisfied. However, if the manifold is non-compact, there may be deformations through incomplete structures. This cannot happen in dimensions greater than 3 (Garland-Raghunathan [13]); but there are always non-trivial deformations in dimension 3 (Thurston [24]) in the non-compact case.In this paper, we extend this rigidity and deformation theory to a class of finite volume, orientable 3-dimensional hyperbolic cone-manifolds, i.e., hyperbolic structures on 3-manifolds with cone-like singularities along a knot or link. Our main result is that such structures are locally rigid if the cone angles are fixed, under the extra hypothesis that all cone angles are at most 21r. We can view the singular structure as an incomplete, smooth structure on the complement of the singular locus whose metric completion is the singular cone structure. The space of deformations of structures on this open manifold has non-zero dimension, so there will be deformations without the condition that the cone angles remain fixed. We show that, it is possible to defQrm the structure so that the metric completionJs still a cone-manifold, and that one

The shape of hyperbolic Dehn surgery space

“…Once the required Hodge theorem is proved, similar arguments to those in [9] imply the following local rigidity and local parametrization result. The previous cone angle restriction has been removed and is replaced by a mild restriction on the tube radius.…”

“…The proof of this result uses the harmonic deformation theory we developed in [9] to deform the finite volume complete hyperbolic structure on the interior of X . The deformation consists of a family of singular hyperbolic metrics on the filled manifolds where the singularities lie along geodesics isotopic to the cores of the attached solid tori.…”

“…Any deformation for which this boundary term goes to zero as the radius of the tube goes to zero is seen to be trivial. The main step in [9] is to show that, for any infinitesimal deformation where all the cone angles are held constant, the boundary term does, indeed, go to zero as long as the cone angles are at most 2 . The fact that the analysis involves arbitrarily small neighborhoods of the singular locus means that it depends on the asymptotic behavior of harmonic deformations near the singular locus.…”

“…The original local rigidity theory in [9] applies only to hyperbolic 3-manifolds with conical singularities along a geodesic link where the cone angles are restricted to be at most 2 . The argument involves finding, for any infinitesimal deformation of the hyperbolic cone structure, a harmonic representative and then utilizing a Weitzenböck formula for such harmonic infinitesimal deformations.…”

“…For the analytic techniques in [9] to work, it is necessary to restrict all the cone angles to be at most 2 . While this is adequate for the Dehn filling problem, there are other situations where it is important to be able to deal with larger cone angles or with a more general type of singularity.…”