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Thurston norms, $L^2$-norms, geodesic laminations, and Lipschitz maps

Xiaolong Hans Han

Abstract

For closed hyperbolic $3$-manifolds $M$ with volume less than a constant $V$, we prove an inequality regarding the geometric $L^2$-norm and the topological Thurston norm, which is qualitatively sharp and verifies a conjecture of Brock and Dunfield in this case. Generically, we show that the $L^2$-norm is less than a constant $c(V)$ times the Thurston norm by showing that any least area closed surface is disjoint from the thin part. We then study the connection between the Thurston norm, best Lipschitz circle-valued maps, and maximal stretch laminations, building on the recent work of Daskalopoulos and Uhlenbeck, and Farre, Landesberg and Minsky. We show that the distance between a level set and its translation is the reciprocal of the Lipschitz constant, bounded by the topological entropy of the pseudo-Anosov monodromy if $M$ fibers. For infinitely many examples constructed by Rudd, we show the entropy is bounded from below by one-third the length of the circumference.

Thurston norms, $L^2$-norms, geodesic laminations, and Lipschitz maps

Abstract

For closed hyperbolic -manifolds with volume less than a constant , we prove an inequality regarding the geometric -norm and the topological Thurston norm, which is qualitatively sharp and verifies a conjecture of Brock and Dunfield in this case. Generically, we show that the -norm is less than a constant times the Thurston norm by showing that any least area closed surface is disjoint from the thin part. We then study the connection between the Thurston norm, best Lipschitz circle-valued maps, and maximal stretch laminations, building on the recent work of Daskalopoulos and Uhlenbeck, and Farre, Landesberg and Minsky. We show that the distance between a level set and its translation is the reciprocal of the Lipschitz constant, bounded by the topological entropy of the pseudo-Anosov monodromy if fibers. For infinitely many examples constructed by Rudd, we show the entropy is bounded from below by one-third the length of the circumference.
Paper Structure (12 sections, 48 theorems, 192 equations, 4 figures)

This paper contains 12 sections, 48 theorems, 192 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Least area surfaces in a Margulis tube
  4. The Thurston norm
  5. The proof of \ref{['theorem main conjecture BD']}
  6. The generic growth and $\epsilon$ homologically thick
  7. The geometry of best Lipschitz maps, geodesic laminations, and fibrations
  8. The Lipschitz constants and best laminations
  9. The Thurston norm, the entropy, and the fiber translation length
  10. Harmonic expansion in a Margulis tube
  11. Formulas, the equation, and the series expansion in terms of hypergeometric functions
  12. The conjecture without volume upper bound

Key Result

Theorem 1.1

Let $V>0$. Then for all closed hyperbolic $3$-manifolds $M$ whose volume $\leq V$ and $\textnormal{inj}(M)\leq 58/10^6$, there exists a constant $c(V)$ such that

Figures (4)

  • Figure 1: Glue invariant annulus $A_i$ to homotope $D_i$ to $\overline{D}_i$
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (101)

  • Theorem 1.1
  • Lemma 1.1
  • Lemma 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 1.5
  • Corollary 1.5
  • ...and 91 more