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Volume of unit balls associated to quadratic differentials

Weixu Su, Shenxing Zhang

TL;DR

This work analyzes the Thurston volume $B(X,q)$ of the unit ball in the measured lamination space $\mathcal{ML}_g$ associated to a holomorphic quadratic differential $q$ on a genus $g$ surface. It develops a degeneration framework via thick–thin decompositions, the Deligne–Mumford compactification, and a collar-lemma for quadratic differentials to obtain sharp lower and upper bounds for $B(X,q)$, and proves that $B$ is not a proper function on the moduli space, while $B(X,q)$ is $p$-integrable for all $0<p<1$ with respect to the Masur–Veech measure. The paper also identifies a precise boundedness criterion: $B(X_n,q_n)$ remains bounded precisely along regular sequences, and diverges otherwise, with the behavior tied to the limiting area distribution among cylinders and thick components. Jenkins–Strebel examples with long cylinders illustrate non-properness and show the sharpness of the degeneration analysis. These results provide a route to understanding random flat surfaces via the growth of simple-closed-curve counts under the flat metric determined by $q$.

Abstract

Associated to a holomorphic quadratic differential is a unit ball of the measured lamination space. The Thurston volume of the unit ball defines a function on the moduli space. We show that the volume function is not proper and characterize when it tends to infinity. We prove that the volume function is $p$-integrable for any $0<p<1$.

Volume of unit balls associated to quadratic differentials

TL;DR

This work analyzes the Thurston volume of the unit ball in the measured lamination space associated to a holomorphic quadratic differential on a genus surface. It develops a degeneration framework via thick–thin decompositions, the Deligne–Mumford compactification, and a collar-lemma for quadratic differentials to obtain sharp lower and upper bounds for , and proves that is not a proper function on the moduli space, while is -integrable for all with respect to the Masur–Veech measure. The paper also identifies a precise boundedness criterion: remains bounded precisely along regular sequences, and diverges otherwise, with the behavior tied to the limiting area distribution among cylinders and thick components. Jenkins–Strebel examples with long cylinders illustrate non-properness and show the sharpness of the degeneration analysis. These results provide a route to understanding random flat surfaces via the growth of simple-closed-curve counts under the flat metric determined by .

Abstract

Associated to a holomorphic quadratic differential is a unit ball of the measured lamination space. The Thurston volume of the unit ball defines a function on the moduli space. We show that the volume function is not proper and characterize when it tends to infinity. We prove that the volume function is -integrable for any .
Paper Structure (16 sections, 10 theorems, 47 equations, 1 figure)

This paper contains 16 sections, 10 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main theorems
  3. Preliminaries
  4. Teichmüller space and moduli space
  5. Measured laminations
  6. Dehn-Thurston coordinates
  7. Lower bounds and limits
  8. An uniform lower bound for $B(X,q)$.
  9. Thick-thin decomposition
  10. Limits of quadratic differentials
  11. Length estimates
  12. Upper bounds and integration
  13. Collar lemma for quadratic differentials
  14. Upper bounds
  15. $p$-integrability
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\left\{(X_n,q_n)\right\}$ be a divergent sequence in $Q^1\mathcal{M}_{g}$. Then $B(X_n, q_n) \to \infty$ except the case that the sequence has a regular subsequence. When the sequence $(X_n,q_n)\in Q^1\mathcal{M}_{g}$ is regular, $B(X_n, q_n)$ are bounded above by a constant depending only on t

Figures (1)

  • Figure 1: In the figure, $\beta_1$ is represented by an arc connecting $x$ to $y$. If $\beta_1$ spirals around $C_j$ more than once, then it must intersect a horizontal leaf of $q_n$ at a point $z$. The union of the horizontal segments $\widehat{zp}$ and $\widehat{py}$ is shorter than the segment of $\beta_1$ from $z$ to $y$.

Theorems & Definitions (24)

  • Definition 1: Volume function
  • Definition 2
  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • ...and 14 more