Schwarzian Bounds on Bending in Hyperbolic 3-Manifolds

Abstract

The Schwarzian derivative provides a classical analytic measure of how far a holomorphic map of the disk is from being M\"obius, 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 $\|\beta_f\|_L$. In particular, for univalent maps with $\|Sf\|_\infty< 1/2$, we show that $\|\beta_f\|_L$ 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 Teichm\"uller 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.