On the continuity of bending

Christos Kourouniotis
1998 The Epstein Birthday Schrift   unpublished
We examine the dependence of the deformation obtained by bending quasi-Fuchsian structures on the bending lamination. We show that when we consider bending quasi-Fuchsian structures on a closed surface, the conditions obtained by Epstein and Marden to relate weak convergence of arbitrary laminations to the convergence of bending cocycles are not necessary. Bending may not be continuous on the set of all measured laminations. However we show that if we restrict our attention to laminations with
more » ... on negative real and imaginary parts then the deformation depends continuously on the lamination. Theorem 3 The mapping ML ++ (S) × Q(S) → T (Q(S)) : (µ, [ρ]) → T µ ([ρ]) is continuous, and holomorphic in [ρ]. We fix a point [ρ] ∈ Q(S), represented by the homomorphism ρ : Γ 0 → P SL(2, C) obtained by conjugation with the quasiconformal homeomorphism φ : C → C. We denote the image of ρ by Γ. The limit set of Γ 0 is R. Then
doi:10.2140/gtm.1998.1.317 fatcat:2andi2e77ba6ji4qzjxpjdmdlu