Limits of Teichmüller maps

Abstract

Let \(\Delta = \{ z \in {\Bbb C}:\left| z \right| 1} ∪ {∞}. A Teichmuller map is a quasiconformal homeomorphism f of ℂ that is conformal outside of Δ* and such that, in Δ, the complex dilatation of f is of the form \(k\overline \varphi \left| \varphi \right|\), where 0 ≤ k < 1 and φ is holomorphic. We consider sequences of such Teichmuller maps {fj} whose complex dilatations are of the form \({k_j}{\overline \varphi _j}\left| {{\varphi _j}} \right|\), where φj are holomorphic mappings, kj → 1, and φj tends to a holomorphic mapping φ uniformly on compact subsets as j → ∞. We assume that the L1-norms of φj and φ are uniformly bounded. If fj are suitably normalized, it is possible to pass to a subsequence such that fj tends to a conformal limit f outside \(\overline \Delta \). Since the fj are not uniformly quasiconformal, such a limit need not exist in \(\overline \Delta \). We show that there exists a subsequence of {fj} which tends to a modified form of a limit, called an extended limit, in \(\overline \Delta \). We construct a subsequence and an extended limit using a partition of \(\overline \Delta \), denoted D, whose elements are closed sets constructed from vertical trajectories of φ as well as some closed arcs and points of ϖΔ. The extended limit, also denoted f, is defined on Δ* ∪ D and satisfies a continuity condition called semicontinuity. The image fD = {f (X): X ∈ D} is a family of closed sets of ℂ which partition ℂ \ fΔ*. The extended limit is a limit of fj’s in a sense which we call semi-convergence. If sets of D are collapsed to points, and similarly f (X), X ∈ D, are collapsed to points, the quotient spaces are homeomorphic to ℂ and f is a homeomorphism between them.