Integral means spectrum functionals on Teichmuller spaces

TL;DR

This work defines and analyzes integral means spectrum (IMS) functionals on Teichmüller spaces, proving their continuity on the closure of the universal Teichmüller space and on the universal asymptotic Teichmüller space, via a Pre-Schwarzian model. It establishes a tight connection between IMS functionals and Bers/Pre-Bers mappings, and shows that the IMS of any univalent function admitting a quasiconformal extension is strictly below the universal IMS for nonzero $t$, implying that global extremals lie beyond the QC class $\mathcal{S}_q$. The results unify spectral extremal questions for univalent functions with QC extensions, clarify continuity structures of IMS on $T$, $\overline{T}_1$, and $AT$, and discuss boundary phenomena, holomorphic motions, and open conjectures about extremals on Teichmüller space boundaries.

Abstract

In this paper we introduce and study the integral means spectrum (IMS) functionals on Teichmüller spaces. We show that the IMS functionals on the closure of the universal Teichmüller space and the universal asymptotic Teichmüller space are both continuous. During the proof, we consider the Pre-Schwarzian derivative model of universal asymptotic Teichmüller space and establish some new results for it. We also show that the integral means spectrum of any univalent function admitting a quasiconformal extension to the extended complex plane is strictly less than the universal integral means spectrum.

Theorems & Definitions (80)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Definition 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 1.10