Metric definition of quasiconformality and exceptional sets

DOI10.1007/S00208-023-02723-6arXiv2111.02918OpenAlexW4386980866MaRDI QIDQ6382217

Dimitrios Ntalampekos

Publication date: 4 November 2021

Abstract: We present a new generalized metric definition of quasiconformality for Euclidean space, requiring that at each point there exists a sequence of uncentered open sets with bounded eccentricity shrinking to that point such that the images also have bounded eccentricity. This generalizes results of Gehring and Heinonen-Koskela on the metric definition of quasiconformality. We also study exceptional sets for this definition, in connection with sets that are negligible for extremal distance. We introduce the class of CNED sets, generalizing the notion of NED sets studied by Ahlfors-Beurling. A set

E

is CNED if the conformal modulus of a curve family is not affected when one restricts to the subfamily intersecting

E

at countably many points. We show as our main theorem that CNED sets are exceptional for the definition of quasiconformality. Our main theorem is an ultimate generalization of results of Gehring, Heinonen-Koskela, and Kallunki-Koskela. We prove that several classes of sets that were known to be exceptional are also CNED, including sets of

s i g m a

-finite Hausdorff

(n - 1)

-measure and boundaries of domains with

n

-integrable quasihyperbolic distance. Thus, this work puts in common framework many known results on the problem of quasiconformal removability and provides evidence that the CNED condition should also be necessary for removability. We prove that countable unions of closed (C)NED sets are (C)NED, and therefore we enlarge significantly the known classes of quasiconformally removable sets. On the other hand, we present examples showing that unions of non-closed negligible or exceptional sets need not be negligible or exceptional, respectively. Finally, we give an application to the problem of rigidity of circle domains.

Full work available at URL: https://doi.org/10.1007/s00208-023-02723-6

Mathematics Subject Classification ID

Quasiconformal mappings in (mathbb{R}^n), other generalizations (30C65) Quasiconformal mappings in the complex plane (30C62) Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions (31A15) Potentials and capacities, extremal length and related notions in higher dimensions (31B15)