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On the maximal dilatation of quasiconformal minimal Lagrangian extensions

  • Andrea Seppi
Original Paper
  • 3 Downloads

Abstract

Given a quasisymmetric homeomorphism \(\varphi \) of the circle, Bonsante and Schlenker proved the existence and uniqueness of the minimal Lagrangian extension \(f_\varphi :\mathbb {H}^2\rightarrow \mathbb {H}^2\) to the hyperbolic plane. By previous work of the author, its maximal dilatation satisfies \(\log K(f_\varphi )\le C||\varphi ||_{cr}\), where \(||\varphi ||_{cr}\) 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.

Keywords

Quasiconformal mappings Universal Teichmüller space Minimal Lagrangian diffeomorphisms Maximal dilatation Cross-ratio norm 

Mathematics Subject Classification

32G15 30F60 53C43 

Notes

Acknowledgements

I am extremely grateful to an anonymous referee for several comments which highly improved the present article. I would like to thank Jean-Marc Schlenker for motivating me towards questions of this type since several years, and Francesco Bonsante and Jun Hu for many interesting discussions on related topics.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.CNRS and Université Grenoble AlpesGiéresFrance

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