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Best Lipschitz maps and Earthquakes

Georgios Daskalopoulos, Karen Uhlenbeck

Abstract

This is the third paper in a series in which we prove Thurston's conjectural duality between best Lipschitz maps and transverse measures. In the second paper we found a special class of best Lipschitz maps between hyperbolic surfaces (infinity harmonic maps), which induce dual Lie algebra valued transverse measures with support on Thurston's canonical lamination. The present paper examines these Lie algebra valued measures in greater detail. For any measured lamination we are led to define a Lie algebra valued measure and conversely every Lie algebra valued transverse measure arrises from this process. Furthermore, we show that such measures are infinitesimal earthquakes. This construction provides a natural correspondence between best Lipschitz maps and earthquakes.

Best Lipschitz maps and Earthquakes

Abstract

This is the third paper in a series in which we prove Thurston's conjectural duality between best Lipschitz maps and transverse measures. In the second paper we found a special class of best Lipschitz maps between hyperbolic surfaces (infinity harmonic maps), which induce dual Lie algebra valued transverse measures with support on Thurston's canonical lamination. The present paper examines these Lie algebra valued measures in greater detail. For any measured lamination we are led to define a Lie algebra valued measure and conversely every Lie algebra valued transverse measure arrises from this process. Furthermore, we show that such measures are infinitesimal earthquakes. This construction provides a natural correspondence between best Lipschitz maps and earthquakes.

Paper Structure

This paper contains 21 sections, 35 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $p$-approximation of infinity harmonic maps
  4. The hyperboloid in $\mathbb R^{2,1}$
  5. Conservation laws
  6. The limiting measures
  7. 1-cocycles and the De Rham isomorphism
  8. Constructing global primitives of $V$ and $W$
  9. Remark on interpreting $v$, $w$ as sections of affine bundles
  10. Lie algebra valued transverse measures
  11. Transverse measures with values in the Lie algebra
  12. The standard Lie algebra valued transverse measure on a measured geodesic lamination
  13. Proof of Theorem \ref{['thmrepresentearth']}
  14. Comment on convergence of laminations
  15. Mass equals length
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $M$, $N$ be closed hyperbolic surfaces and let $u:M \rightarrow N$ be an infinity harmonic map such that $u=\lim_{p \rightarrow \infty} u_p$ for a sequence $u_p$ of $J_p$-minimizers in the same homotopy class. Then after passing to a subsequence and after normalizing by positive constants $\kapp

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • Proposition 2.4
  • ...and 57 more