On the maximal dilatation of quasiconformal minimal Lagrangian extensions

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.

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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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Correspondence to Andrea Seppi.

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Seppi, A. On the maximal dilatation of quasiconformal minimal Lagrangian extensions. Geom Dedicata 203, 25–52 (2019). https://doi.org/10.1007/s10711-019-00422-8

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Keywords

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

Mathematics Subject Classification

  • 32G15
  • 30F60
  • 53C43