Some remarks on bounded earthquakes

TL;DR

This paper investigates the behavior of earthquakes on hyperbolic surfaces, establishing conditions for asymptotic conformality, continuity of earthquake paths, and triviality of measured laminations at infinity.

Contribution

It provides new criteria linking measured laminations and conformal changes, and proves continuity properties of earthquake paths in Teichmüller space.

Findings

Asymptotically conformal change occurs iff the lamination is asymptotically trivial.

Earthquake path contraction is continuous in Teichmüller space.

Measured lamination vanishing at high rate must be trivial.

Abstract

We first show that an earthquake of a geometrically infinite hyperbolic surface induces an asymptotically conformal change in the hyperbolic metric if and only if the measured lamination associated with the earthquake is asymptotically trivial on the surface. Then we show that the contraction along earthquake paths is continuous in the Teichm\"uller space of any hyperbolic surface. Finally, we show that if a measured lamination vanishes while approaching infinity at the rate higher than the distance to the boundary then it must be trivial.