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 log K(fφ) ≤ C| | φ| | cr, where | | φ| | cr 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.
On the maximal dilatation of quasiconformal minimal Lagrangian extensions
Seppi A.
2019-01-01
Abstract
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 log K(fφ) ≤ C| | φ| | cr, where | | φ| | cr 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.
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