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Is there always an extremal Teichmüller mapping?

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Abstract

Given a quasisymmetric self-homeomorphismh of the unit circleS 1, letQ(h) be the set of all quasiconformal mappings with the boundary correspondenceh. In [1], it was shown that there existsh for which no extremal extension inQ(h) as a Teichmüller mapping is possible. This disproved some conjectures of long standing. In the example constructed there, the boundary correspondence has a single extremal quasiconformal extension. We show that even when there are infinitely many extremal extensions of the boundary values, it may still happen that none of the extensions is a Teichmüller mapping. An infinitesimal version of this result is also obtained.

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Authors and Affiliations

  1. Institute of Mathematics Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100080, Beijing, People's Republic of China

    Guowu Yao

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  1. Guowu Yao
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Correspondence to Guowu Yao.

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The research was supported by the China Postdoctoral Science Foundation and by the National Natural Science Foundation of China (Grant No. 10401036).

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Yao, G. Is there always an extremal Teichmüller mapping?. J. Anal. Math. 94, 363–375 (2004). https://doi.org/10.1007/BF02789054

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  • DOI: https://doi.org/10.1007/BF02789054

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