Schwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volume

Abstract

To a complex projective structure Σ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms φ Σ ∞ and φ Σ 2 of the quadratic differential φ Σ of Σ given by the Schwarzian derivative of the associated locally univalent map. We show that these give a unifying approach that generalizes a number of important, well known results for convex cocompact hyperbolic structures on 3-manifolds, including bounds on the Lipschitz constant for the nearest-point retraction and the length of the bending lamination. We then use these bounds to begin a study of the Weil-Petersson gradient flow of renormalized volume on the space CC(N) of convex cocompact hyperbolic structures on a compact manifold N with incompressible boundary, leading to a proof of the conjecture that the renormalized volume has infimum given by one-half the simplicial volume of DN, the double of N. 1. the retract map ∂ c M → ∂C(M) is 1 + 2 φ M ∞ -Lipschitz, and 2. L(λ M ) ≤ 4π|χ(∂ c M)| φ M ∞ where L(λ M ) is the length of the bending lamination λ M of ∂C(M).

Measured foliations at infinity of quasi-Fuchsian manifolds near the Fuchsian locus

“…measured lamination λ) the derivatives d(ext(F + )), d(l(λ + )) : T T (S) → R are considered as elements in the cotangent space T * T (S). Moreover, we also have the upper bound from [7] that l m± (λ ± ) ≤ 6π|χ(S)| whereas, from [37] we have similar upper bounds on the extremal length ext [c±] (F ± ) to be 3π|χ(S)|, where χ(S) is the Euler characterisitic of S.…”

“…where W (2,2) is the classical Sobolev space. It is clear that u s solves Equation ( 5) if and only if (u s , s) is a solution for Equation (7). We want to first compute the differential dF (u, s) and the linearised operator with respect to s of the function F (u s , s).…”

The dual Bonahon–Schläfli formula

“…We remind that the mean curvature here is the trace of the shape operator B, which is defined using the exterior normal vector field to ∂ N; this explains why the relation above differ by a factor −2 from the one in [BBB19]. In particular, the proof of [BBB19, Proposition 3.4] shows also:…”

A bound on the $L^2$-norm of a projective structure by the length of the bending lamination

“…Theorem 1.1 (Anderson,[BBB,Theorem 2.10]) If Σ is a projective structure with Schwarzian parameterization (X Σ , Φ Σ ) and Thurston parameterization (Y Σ , λ Σ ) then…”