Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maximal cusps are not dense

Ryo Matsuda

Abstract

We proved that the Maximal cusp is not dense on the Bers boundary of the Teichmüller space of infinite type Riemann surfaces satisfying some analytic conditions. This is a counterexample to the infinite-type case of the McMullen result for finite-type Riemann surfaces. More precisely, we showed that maximal cusps cannot approach the points on the Bers boundary corresponding to the deformation by the David map, which can be regarded as a degenerate quasiconformal map in the neighborhood of one end. In addition, to prove this, we used quasiconformal deformations in the neighborhood of a fixed end. We then proved that such a subset of the Teichmüller space has a manifold structure.

Maximal cusps are not dense

Abstract

We proved that the Maximal cusp is not dense on the Bers boundary of the Teichmüller space of infinite type Riemann surfaces satisfying some analytic conditions. This is a counterexample to the infinite-type case of the McMullen result for finite-type Riemann surfaces. More precisely, we showed that maximal cusps cannot approach the points on the Bers boundary corresponding to the deformation by the David map, which can be regarded as a degenerate quasiconformal map in the neighborhood of one end. In addition, to prove this, we used quasiconformal deformations in the neighborhood of a fixed end. We then proved that such a subset of the Teichmüller space has a manifold structure.

Paper Structure

This paper contains 11 sections, 33 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasihyperbolic metric
  4. Kleinian groups and Bers embedding
  5. The collar lemma and Wolpert--Fujikawa's inequality
  6. Bergman kernel
  7. Maximal cusps are not dense
  8. Partial deformation at end and Bers embedding
  9. The restriction
  10. Bers slice
  11. Continued proof of Theorem \ref{['main theorem']}

Key Result

Theorem 1

Theorems & Definitions (59)

  • Theorem : Theorem \ref{['main theorem']}, Maximal cusps are not dense
  • Theorem : Theorem \ref{['partial deformation at ends']} & Theorem \ref{['thm: surj. par deg def']}, Teichmüller space of partial quasiconformal deforming end
  • Proposition 1
  • Theorem 2: Gehring--Osgood, GO
  • Remark 3
  • Theorem 4: Bers embedding (G, 5.6, Theorem 4)
  • Remark 5
  • Remark 6
  • Theorem 7: BersB, MaskitMas
  • Remark 8
  • ...and 49 more