A cyclic extension of the earthquake flow II

TL;DR

This paper extends properties of the earthquake flow to the landslide flow on Teichmüller space, revealing its Hamiltonian nature, properness, surjectivity, symplecticity, fixed points, and boundary extension, with applications to hyperbolic geometry.

Contribution

It introduces new properties of the landslide flow, including its Hamiltonian structure, properness, surjectivity, symplecticity, and boundary extension, advancing understanding of Teichmüller space dynamics.

Findings

Landslide flow shares key properties with earthquake flow.

The smooth grafting map is proper, surjective, and symplectic.

The composition of two landslides has a fixed point on Teichmüller space.

Abstract

The landslide flow, introduced in [5], is a smoother analog of the earthquake flow on Teichm\"uller space which shares some of its key properties. We show here that further properties of earthquakes apply to landslides. The landslide flow is the Hamiltonian flow of a convex function. The smooth grafting map $sgr$ taking values in Teichm\"uller space, which is to landslides as grafting is to earthquakes, is proper and surjective with respect to either of its variables. The smooth grafting map $SGr$ taking values in the space of complex projective structures is symplectic (up to a multiplicative constant). The composition of two landslides has a fixed point on Teichm\"uller space. As a consequence we obtain new results on constant Gauss curvature surfaces in 3-dimensional hyperbolic or AdS manifolds. We also show that the landslide flow has a satisfactory extension to the boundary of Teichm\"uller space.