Uniform degeneration of hyperbolic surfaces with boundary along harmonic map rays

TL;DR

This paper investigates how hyperbolic surfaces degenerate along harmonic map rays, showing convergence to an -tree and characterizing limits in the Thurston boundary, with implications for Teichmfcller theory.

Contribution

It establishes uniform convergence of rescaled distance functions to intersection numbers with measured foliations, linking harmonic map rays to -trees and Thurston boundary limits.

Findings

Rescaled distance functions converge uniformly to intersection numbers.

Hyperbolic surfaces along the ray converge to the dual -tree.

Limits in the Thurston boundary are characterized explicitly.

Abstract

We study the degeneration of hyperbolic surfaces along a ray given by the harmonic map parametrization of Teichm\"uller space. The direction of the ray is determined by a holomorphic quadratic differential on a punctured Riemann surface, which has poles of order $\geq 2$ at each puncture. We show that the rescaled distance functions of the universal covers of hyperbolic surfaces uniformly converge, on a certain non-compact region containing a fundamental domain, to the intersection number with the vertical measured foliation given by the holomorphic quadratic differential determining the direction of the ray. This implies that hyperbolic surfaces along the ray converge to the dual $\mathbb{R}$-tree of the vertical measured foliation in the sense of Gromov-Hausdorff. As an application, we determine the limit of the hyperbolic surfaces in the Thurston boundary.