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A comparison between the Avila-Gouëzel-Yoccoz norm and the Teichmüller norm

Weixu Su, Shenxing Zhang

Abstract

We give a comparison between the Avila-Gouëzel-Yoccoz norm and the Teichmüller norm on the principal stratum of holomorphic quadratic differentials.

A comparison between the Avila-Gouëzel-Yoccoz norm and the Teichmüller norm

Abstract

We give a comparison between the Avila-Gouëzel-Yoccoz norm and the Teichmüller norm on the principal stratum of holomorphic quadratic differentials.
Paper Structure (10 sections, 6 theorems, 43 equations, 2 figures)

This paper contains 10 sections, 6 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The moduli space of quadratic differentials
  4. Canonical double cover
  5. The AGY norm.
  6. The upper bound
  7. Delaunay triangulation
  8. The proof of upper bound
  9. The order $\frac{1}{r}$ in Proposition \ref{['prop:upper']} is sharp.
  10. The lower bound

Key Result

Theorem 1.1

Let $(X,q)\in \mathcal{P}_g$ with area $\|q\|=1$. Let $\rho: \hat{X}\to X$ be the canonical double cover such that $\rho^*q=\omega^2$. Then for any $\eta \in H_{-1}^{0,1}(\hat{X})$, we have where $2 r$ is the shortest length of saddle connections on $(\hat{X}, \omega)$.

Figures (2)

  • Figure 1: The examples of Kahn-Wright KW21.
  • Figure :

Theorems & Definitions (12)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 2.1
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Theorem 4.1
  • proof : The lower bound in Theorem \ref{['thm:main']}
  • ...and 2 more