Limits of Teichmüller maps

Abstract

Let $\Delta = \{ z \in {\Bbb C}:\left| z \right| < 1\} ,\overline \Delta = \{ z \in {\Bbb C}:\left| z \right| \leqslant 1\}$, and ℂ̄ \ Δ̄ = {z ∈ ℂ :|z| > 1} ∪ {∞}. A Teichmüller map is a quasiconformal homeomorphism f of ℂ̄ 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 Teichmüller maps {f _j } whose complex dilatations are of the form ${k_j}{\overline \varphi _j}\left| {{\varphi _j}} \right|$, where φ _j are holomorphic mappings, k _j → 1, and φ _j tends to a holomorphic mapping φ uniformly on compact subsets as j → ∞. We assume that the L ¹-norms of φ _j and φ are uniformly bounded. If f _j are suitably normalized, it is possible to pass to a subsequence such that f _j tends to a conformal limit f outside $\overline \Delta$. Since the f _j are not uniformly quasiconformal, such a limit need not exist in $\overline \Delta$. We show that there exists a subsequence of {f _j } 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 f D = {f (X): X ∈ D} is a family of closed sets of ℂ̄ which partition ℂ̄ \ fΔ*. The extended limit is a limit of f _j ’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 ℂ̄ and f is a homeomorphism between them.