Uniform degeneration of hyperbolic surfaces with boundary along harmonic map rays

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.