Uniform degeneration of hyperbolic surfaces with boundary along harmonic map rays

TL;DR

The paper analyzes how hyperbolic surfaces with boundary degenerate along harmonic-map rays directed by meromorphic quadratic differentials with poles of order at least two. It proves that, after rescaling, distance functions on the universal cover converge uniformly on non-compact regions to the vertical foliation’s intersection function, yielding a Gromov–Hausdorff limit to the dual $\mathbb{R}$-tree. This framework provides an explicit description of the Thurston boundary limit for rays in Teichmüller space with boundaries or crown ends. The results extend classical degeneration phenomena to surfaces with boundary, explain the limiting metric geometry via collapsing foliations, and tie the analytic data of principal parts to asymptotic geometric limits in the Thurston compactification.

Abstract

We study the degeneration of hyperbolic surfaces along a ray given by the harmonic map parametrization of Teichmüller 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.

Figures (11)

• Figure 1: Hyperbolic surface with a closed boundary (left side) and a crown end (right side).
• Figure 2: Exceptional surfaces
• Figure 3: The blue colored domain is $U$. The dotted lines denote the vertical lines bounding maximal half-plane domains.
• Figure 4: The dotted lines denote the vertical lines of $\varphi$. The thick line denotes the boundary of $P$.
• Figure 5: This illustrates the steps of the deformation around zeros. The dotted lines are leaves of the vertical foliation, the solid lines are leaves of the horizontal foliation. The upper left side: Step 1, the upper right side: Step 2, the lower left side: Step 3, the lower right side: Step 4.

Theorems & Definitions (44)

• Theorem 1: \ref{['corollary:GromovHausdorff']}
• Theorem 2: \ref{['theorem:GeneralCases']}
• Corollary 3: \ref{['corollary:PointInBoundary']}
• Definition 2.1: analytic residue
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Theorem 2.5: Existence, wolf1991infinitegupta2021harmonicallegretti2021stability
• Theorem 2.6: Parametrization, gupta2021harmonicsagman2023infiniteallegretti2021stability
• Proposition 2.7: minsky1992harmonic