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

TL;DR

This paper establishes an upper bound on the $L^2$-norm of the holomorphic quadratic differential associated with a complex projective structure, in terms of the length of its bending lamination, using $W$-volume theory.

Contribution

It provides a new quantitative relationship between the $L^2$-norm of the quadratic differential and the bending lamination length, utilizing Krasnov-Schlenker's $W$-volume theory.

Findings

Upper bounds on the $L^2$-norm of $\

Connection between quadratic differential norm and lamination length

Application of $W$-volume theory to projective structures

Abstract

One can associate to a complex projective structure on a surface holomorphic quadratic differential $\Phi$ via the Schwarzian derivative and a bending lamination $\lambda$ via the Thurston parameterization. In this note we obtain upper bounds on the $L^2$-norm of $\Phi$ in terms of the length of $\lambda$. The proof uses the theory of $W$-volume introduced by Krasnov-Schlenker.