On the maximal dilatation of quasiconformal minimal Lagrangian extensions

TL;DR

This paper investigates the bounds on the maximal dilatation of minimal Lagrangian extensions of quasisymmetric circle homeomorphisms, providing constraints on the optimal constant relating dilatation and cross-ratio norm.

Contribution

It establishes constraints on the optimal constant linking maximal dilatation and cross-ratio norm for minimal Lagrangian extensions, and explores specific families of such extensions.

Findings

Derived bounds on the optimal constant C for dilatation-norm inequality.

Analyzed two one-parameter families of minimal Lagrangian extensions.

Discussed potential lower bounds for the dilatation in terms of cross-ratio norm.

Abstract

Given a quasisymmetric homeomorphism $\varphi$ of the circle, Bonsante and Schlenker proved the existence and uniqueness of the minimal Lagrangian extension $f_\varphi:\mathbb{H}^2\to\mathbb{H}^2$ to the hyperbolic plane. By previous work of the author, its maximal dilatation satisfies $\log K(f_\varphi)\leq C||\varphi||$, where $||\varphi||$ denotes the cross-ratio norm. We give constraints on the value of an optimal such constant $C$, and discuss possible lower inequalities, by studying two one-parameter families of minimal Lagrangian extensions in terms of maximal dilatation and cross-ratio norm.