Ray structures on Teichmüller Space

TL;DR

The work introduces and develops harmonic map rays as a unifying conduit between Teichmüller and Thurston geometries on Teichmüller space. By proving energy-minimization limits of harmonic maps yield unique harmonic stretch lines between any pair of hyperbolic surfaces, the authors define a canonical Thurston geodesic and an exponential-like map toward the Thurston boundary. The theory extends from rays to disks, linking Hopf-differential disks to stretch-earthquake disks and Teichmüller disks, and provides compactness, Jenkins–Serrin-type uniqueness, and train-track–tree tools to analyze degenerate limits. The results give new canonical geodesics in the Thurston metric, establish two Thurston geodesic flows, and supply geometric realizations via minimal graphs and crowned surfaces, with implications for boundary behavior and dynamics on moduli space. Overall, the paper furnishes a cohesive energy-driven bridge among three foundational ray structures in Teichmüller theory and advances understanding of their interactions, extensions to surfaces with boundary, and disk-level dynamics toward Thurston boundary points.

Abstract

While there may be many Thurston metric geodesics between a pair of points in Teichmüller space, we find that by imposing an additional energy minimization constraint on the geodesics, thought of as limits of harmonic map rays, we select a unique Thurston geodesic through those points. Extending the target surface to the Thurston boundary yields, for each point $Y$ in Teichmüller space, an \enquote{exponential map} of rays from that point $Y$ onto Teichmüller space with visual boundary the Thurston boundary of Teichmüller space. We first depict harmonic map ray structures on Teichmüller space as a geometric transition between Teichmüller ray structures and Thurston geodesic ray structures. In particular, by appropriately degenerating the source of a harmonic map between hyperbolic surfaces (along \enquote{harmonic map dual rays}), the harmonic map rays through the target converge to a Thurston geodesic; by appropriately degenerating the target of the harmonic map, those harmonic map dual rays through the domain converge to Teichmüller geodesics. We then extend this transition to one from Teichmüller disks through Hopf differential disks to stretch-earthquake disks. These results apply to surfaces with boundary, resolving a question on stretch maps between such surfaces.

Theorems & Definitions (186)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: piecewise harmonic stretch map
• Theorem 1.6: Uniqueness of harmonic stretch lines
• Remark 1.7
• Remark 1.8
• Remark 1.9
• Theorem 1.10