The Schwarzian derivative and measured laminations on Riemann surfaces

Abstract

Abstract. A holomorphic quadratic differential on a hyperbolic Riemann surface has an associated measured foliation, which can be straightened to yield a measured geodesic lamination. On the other hand, a quadratic differential can be considered as the Schwarzian derivative of a È 1 structure, to which one can naturally associate another measured geodesic lamination using grafting.We compare these two relationships between quadratic differentials and measured geodesic laminations, each of which yields a homeomorphism ML (S) → Q(X) for each conformal structure X on a compact surface S. We show that these maps are nearly the same, differing by a multiplicative factor of −2 and an error term of lower order than the maps themselves (which we bound explicitly).As an application we show that the Schwarzian derivative of a È 1 structure with Fuchsian holonomy is close to a 2π-integral JenkinsStrebel differential. We also study compactifications of the space of È 1 structures using the Schwarzian derivative and grafting coordinates; we show that the natural map between these extends to the boundary of each fiber over Teichmüller space, and we describe this extension.

Riemann surfaces with boundary and natural triangulations of the Teichmüller space

“…Finally, we show that, for large r, two results of Dumas [Dum06], [Dum07b] for compact surfaces still hold (Theorem 5.12). The first one says that, for r large, the Strebel differential ϕ is well-approximated in L 1 loc (Ṙ) by the Hopf differential of the collapsing map associated to P(Ṙ, rp), that is the quadratic differential which is dz 2 on the flat cylinders S 1 × [0, ∞) and is zero on the hyperbolic part.…”

“…Assertion (b) is also basically proven in [Dum07b] up to minor considerations: • Let ρ be the hyperbolic metric onṘ and ρ m the Thurston metric on Gr ∞ ( m ) ∼ =Ṙ ; moreover, let β m be the Schwarzian tensor…”

“…and all restrictions to K of metrics in N are pairwise Hölder equivalent with factor and exponent dependent onṘ only (same proof as in Theorem 9.2 of [Dum07b]). …”

“…At the end of Section 5, we investigate the degeneration as r → ∞ of the projective structure induced by the infinite grafting procedure, adapting arguments of Dumas [Dum06], [Dum07b].…”

“…We stress that the proofs of these last results and of the local injectivity of the grafting map assume the knowledge of the papers [SW02], [Dum06] and [Dum07b].…”