Length functions in Teichm\"uller and anti de Sitter geometry
Filippo Mazzoli, Gabriele Viaggi
TL;DR
This paper explores the relationship between length functions in Teichmüller space and anti de Sitter geometry, providing new proofs and coordinates that deepen understanding of these geometric structures.
Contribution
It introduces anti de Sitter-based proofs of key Teichmüller theory results and develops shear-bend coordinates for anti de Sitter 3-manifolds.
Findings
Convexity of length functions along shear paths
Geometric bounds on second variation along earthquakes
Shear-bend coordinates for anti de Sitter manifolds
Abstract
We establish a link between the behavior of length functions on Teichm\"uller space and the geometry of certain anti de Sitter 3-manifolds. As an application, we give new purely anti de Sitter proofs of results of Teichm\"uller theory such as (strict) convexity of length functions along shear paths and geometric bounds on their second variation along earthquakes. Along the way, we provide shear-bend coordinates for Mess' anti de Sitter 3-manifolds.
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Taxonomy
TopicsGeometric and Algebraic Topology · Analytic and geometric function theory · Geometric Analysis and Curvature Flows