On the maximal dilatation of quasiconformal minimal Lagrangian extensions

Andrea Seppi (1)

1 IF - Institut Fourier

Given a quasisymmetric homeomorphism ϕ of the circle, Bonsante and Schlenker proved the existence and uniqueness of the minimal Lagrangian extension f_ϕ : H^2 → H^2 to the hyperbolic plane. By previous work of the author, its maximal dilatation satisfies log K(f_ϕ) ≤ 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.

Domaines: Variables complexes [math.CV]; Analyse classique [math.CA]; Géométrie différentielle [math.DG]

Andrea Seppi. On the maximal dilatation of quasiconformal minimal Lagrangian extensions. Geometriae Dedicata, 2019, 203, pp.25-52. ⟨10.1007/s10711-019-00422-8⟩. ⟨hal-01675476v2⟩