Quasiconformal extension for harmonic mappings on finitely connected domains

Iason Efraimidis

doi:10.5802/crmath.233

Analyse et géométrie complexes

Quasiconformal extension for harmonic mappings on finitely connected domains

Iason Efraimidis ¹

¹ Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, TX 79409, United States.

Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 905-909.

Résumé

We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal extension to the whole plane if its Schwarzian derivative is small. We also make the observation that a univalence criterion for harmonic mappings holds on uniform domains.

Reçu le : 2021-02-04
Accepté le : 2021-06-06
Publié le : 2021-09-17

Zbl

DOI : 10.5802/crmath.233

Classification : 30C55, 30C62, 31A05

Affiliations des auteurs :

Iason Efraimidis ¹

¹ Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, TX 79409, United States.

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2021__359_7_905_0,
     author = {Iason Efraimidis},
     title = {Quasiconformal extension for harmonic mappings on finitely connected domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {905--909},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {7},
     year = {2021},
     doi = {10.5802/crmath.233},
     zbl = {07398742},
     language = {en},
}

TY  - JOUR
AU  - Iason Efraimidis
TI  - Quasiconformal extension for harmonic mappings on finitely connected domains
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 905
EP  - 909
VL  - 359
IS  - 7
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.233
LA  - en
ID  - CRMATH_2021__359_7_905_0
ER  -

%0 Journal Article
%A Iason Efraimidis
%T Quasiconformal extension for harmonic mappings on finitely connected domains
%J Comptes Rendus. Mathématique
%D 2021
%P 905-909
%V 359
%N 7
%I Académie des sciences, Paris
%R 10.5802/crmath.233
%G en
%F CRMATH_2021__359_7_905_0

Iason Efraimidis. Quasiconformal extension for harmonic mappings on finitely connected domains. Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 905-909. doi : 10.5802/crmath.233. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.233/

Bibliographie
Cité par

[1] Lars V. Ahlfors Quasiconformal reflections, Acta Math., Volume 109 (1963), pp. 291-301 | DOI | MR | Zbl

[2] Kari Astala; Juha Heinonen On quasiconformal rigidity in space and plane, Ann. Acad. Sci. Fenn., Math., Volume 13 (1988) no. 1, pp. 81-92 | DOI | MR | Zbl

[3] Martin Chuaqui; Peter L. Duren; Brad G. Osgood The Schwarzian derivative for harmonic mappings, J. Anal. Math., Volume 91 (2003), pp. 329-351 | DOI | MR | Zbl

[4] Peter L. Duren Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, 156, Cambridge University Press, 2004 | MR | Zbl

[5] Iason Efraimidis Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains (https://arxiv.org/abs/2009.14766, to appear in the Annales Fennici Mathematici) | Zbl

[6] Frederick W. Gehring; Kari Hag The ubiquitous quasidisk, Mathematical Surveys and Monographs, 184, American Mathematical Society, 2012 | MR | Zbl

[7] Frederick W. Gehring; Brad G. Osgood Uniform domains and the quasihyperbolic metric, J. Anal. Math., Volume 36 (1979), pp. 50-74 | DOI | MR | Zbl

[8] Rodrigo Hernández; María J. Martín Criteria for univalence and quasiconformal extension of harmonic mappings in terms of the Schwarzian derivative, Arch. Math., Volume 104 (2015) no. 1, pp. 53-59 | DOI | MR | Zbl

[9] Rodrigo Hernández; María J. Martín Pre-Schwarzian and Schwarzian derivatives of harmonic mappings, J. Geom. Anal., Volume 25 (2015) no. 1, pp. 64-91 | DOI | MR | Zbl

[10] Olli E. Lehto; K. I. Virtanen Quasiconformal mappings in the plane, Grundlehren der Mathematischen Wissenschaften, 126, Springer, 1973 | MR | Zbl

[11] Olli Martio; Jukka Sarvas Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn., Math., Volume 4 (1979) no. 2, pp. 383-401 | DOI | MR | Zbl

[12] Brad G. Osgood Univalence criteria in multiply-connected domains, Trans. Am. Math. Soc., Volume 260 (1980) no. 2, pp. 459-473 | DOI | MR | Zbl

[13] George Springer Fredholm eigenvalues and quasiconformal mapping, Acta Math., Volume 111 (1964), pp. 121-142 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique