Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 541-556

SYMMETRIZATION AND EXTENSION OF PLANAR BI-LIPSCHITZ MAPS

Leonid V. Kovalev

Syracuse University, Mathematics Department
215 Carnegie, Syracuse, NY 13244, U.S.A.; lvkovale 'at' syr.edu

Abstract. We show that every centrally symmetric bi-Lipschitz embedding of the circle into the plane can be extended to a global bi-Lipschitz map of the plane with linear bounds on the distortion. This answers a question of Daneri and Pratelli in the special case of centrally symmetric maps. For general bi-Lipschitz embeddings our distortion bound has a combination of linear and cubic growth, which improves on the prior results. The proof involves a symmetrization result for bi-Lipschitz maps which may be of independent interest.

2010 Mathematics Subject Classification: Primary 26B35; Secondary 30C35, 31A15.

Key words: Bi-Lipschitz extension, conformal map, harmonic measure.

Reference to this article: L. V. Kovalev: Symmetrization and extension of planar bi-Lipschitz maps. Ann. Acad. Sci. Fenn. Math. 43 (2018), 541-556.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4335