Given a quasisymmetric homeomorphism φ of the circle, Bonsante and
Schlenker proved the existence and uniqueness of the minimal Lagrangian
extension fφ:H2→H2 to the hyperbolic plane. By
previous work of the author, its maximal dilatation satisfies logK(fφ)≤C∣∣φ∣∣, where ∣∣φ∣∣ 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.Comment: 25 pages. Results of Theorem A improved. Several mistakes corrected,
Remark 4.9 added, general exposition improve