Local rigidity of 3-dimensional cone-manifolds

Abstract

We investigate the local deformation space of 3-dimensional conemanifold structures of constant curvature κ ∈ {−1, 0, 1} and coneangles ≤ π. Under this assumption on the cone-angles the singular locus will be a trivalent graph. In the hyperbolic and the spherical case our main result is a vanishing theorem for the first L 2 -cohomology group of the smooth part of the cone-manifold with coefficients in the flat bundle of infinitesimal isometries. We conclude local rigidity from this. In the Euclidean case we prove that the first L 2 -cohomology group of the smooth part with coefficients in the flat tangent bundle is represented by parallel forms.

Local coordinates for \operatorname{SL}(n,\mathbf{C})-character varieties of finite-volume hyperbolic 3-manifolds

“…Proof. When ϑ = dx ⊗ h j + , d ϑ (γ) = h j + , by (8). Therefore ρ n,ε (γ) = (Id +ε h j + )ρ n (γ) is upper triangular with 1 on the diagonal.…”

Deforming Euclidean cone 3–manifolds

“…This remark can be easily proved adapting the arguments of the proof of hyperbolic Dehn filling for orbifolds in Boileau-Porti [3, Appendix B] and using the infinitesimal rigidity results established in Weiss [14].…”

“…The latter assumption may be viewed as a nondegeneracy condition and is in that respect similar to the assumption of not being Seifert fibered in the deformation theory of spherical cone manifolds, cf Boileau et al [2] and Weiss [14].…”

“…Since S U.2/ is compact (resp. the action of SL 2 ‫/ރ.‬ is proper on the irreducible part of R.M; SL 2 ‫,//ރ.‬ cf [14,Lemma 6.24]) and the tangent space to the orbit is given by Proof Let us digress into a more general situation first.…”

“…Our starting point is the following theorem of [14], where the hypothesis about cone angles is used: Theorem 3.7 If M is the smooth part of a closed Euclidean cone manifold with cone angles Ä , then…”