Schwarzian Bounds on Bending in Hyperbolic 3-Manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912625072275456 |
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| author | Bridgeman, Martin Tee, Ming Hong |
| author_facet | Bridgeman, Martin Tee, Ming Hong |
| contents | The Schwarzian derivative provides a classical analytic measure of how far a holomorphic map of the disk is from being Möbius, with Nehari's bounds giving sharp criteria for univalence. Independently, Thurston introduced a geometric parametrization of locally univalent maps via bending measured laminations on the hyperbolic plane, capturing deviation from roundness in hyperbolic three-space. While both approaches quantify the same phenomenon, their precise relationship has remained only implicit. In this paper we establish explicit quantitative bounds relating the Schwarzian norm $\|Sf\|_\infty$ and the bending norm $\|β_f\|_L$. In particular, for univalent maps with $\|Sf\|_\infty< 1/2$, we show that $\|β_f\|_L$ is controlled by an elementary function $B_L(\|Sf\|_L)$ that we compute explicitly. As an application, we obtain new effective bounds on the bending laminations of quasifuchsian manifolds in terms of the Teichmüller distance between their conformal boundary components. Our results sharpen the analytic-geometric correspondence between the Schwarzian derivative and hyperbolic geometry showing that just as small Schwarzian norm forces injectivity, it also forces controlled bending of convex hull boundaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12493 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schwarzian Bounds on Bending in Hyperbolic 3-Manifolds Bridgeman, Martin Tee, Ming Hong Geometric Topology Differential Geometry 30F60, 30F40 The Schwarzian derivative provides a classical analytic measure of how far a holomorphic map of the disk is from being Möbius, with Nehari's bounds giving sharp criteria for univalence. Independently, Thurston introduced a geometric parametrization of locally univalent maps via bending measured laminations on the hyperbolic plane, capturing deviation from roundness in hyperbolic three-space. While both approaches quantify the same phenomenon, their precise relationship has remained only implicit. In this paper we establish explicit quantitative bounds relating the Schwarzian norm $\|Sf\|_\infty$ and the bending norm $\|β_f\|_L$. In particular, for univalent maps with $\|Sf\|_\infty< 1/2$, we show that $\|β_f\|_L$ is controlled by an elementary function $B_L(\|Sf\|_L)$ that we compute explicitly. As an application, we obtain new effective bounds on the bending laminations of quasifuchsian manifolds in terms of the Teichmüller distance between their conformal boundary components. Our results sharpen the analytic-geometric correspondence between the Schwarzian derivative and hyperbolic geometry showing that just as small Schwarzian norm forces injectivity, it also forces controlled bending of convex hull boundaries. |
| title | Schwarzian Bounds on Bending in Hyperbolic 3-Manifolds |
| topic | Geometric Topology Differential Geometry 30F60, 30F40 |
| url | https://arxiv.org/abs/2509.12493 |