The branch set of a quasiregular mapping between metric manifolds

Chang-Yu Guo; Marshall Williams

doi:10.1016/j.crma.2015.10.022

Complex analysis/Topology

The branch set of a quasiregular mapping between metric manifolds
[L'ensemble de branchement d'une application quasi régulière entre variétés métriques]

Présenté par : the Editorial Board

Chang-Yu Guo ¹ ; Marshall Williams ²

¹ Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland
² Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA

Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 155-159.

Résumé
Abstract

Dans cette note, nous annonçons de nouveaux résultats quant à la porosité dénombrable quantitative de l'ensemble des branchements d'une application quasi régulière dans un cadre très général d'espaces métriques. Comme applications de nos résultats, nous répondons à une conjecture récente de Fässler et al., à un problème ouvert de Heinonen–Rickman et à une question ouverte de Heinonen–Semmes.

In this note, we announce some new results on quantitative countable porosity of the branch set of a quasiregular mapping in very general metric spaces. As applications, we solve a recent conjecture of Fässler et al., an open problem of Heinonen–Rickman, and an open question of Heinonen–Semmes.

Reçu le : 2015-08-31
Accepté le : 2015-10-27
Publié le : 2016-01-07

DOI : 10.1016/j.crma.2015.10.022

Affiliations des auteurs :

Chang-Yu Guo ¹ ; Marshall Williams ²

¹ Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland
² Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA

@article{CRMATH_2016__354_2_155_0,
     author = {Chang-Yu Guo and Marshall Williams},
     title = {The branch set of a quasiregular mapping between metric manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {155--159},
     publisher = {Elsevier},
     volume = {354},
     number = {2},
     year = {2016},
     doi = {10.1016/j.crma.2015.10.022},
     language = {en},
}

TY  - JOUR
AU  - Chang-Yu Guo
AU  - Marshall Williams
TI  - The branch set of a quasiregular mapping between metric manifolds
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 155
EP  - 159
VL  - 354
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2015.10.022
LA  - en
ID  - CRMATH_2016__354_2_155_0
ER  -

%0 Journal Article
%A Chang-Yu Guo
%A Marshall Williams
%T The branch set of a quasiregular mapping between metric manifolds
%J Comptes Rendus. Mathématique
%D 2016
%P 155-159
%V 354
%N 2
%I Elsevier
%R 10.1016/j.crma.2015.10.022
%G en
%F CRMATH_2016__354_2_155_0

Chang-Yu Guo; Marshall Williams. The branch set of a quasiregular mapping between metric manifolds. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 155-159. doi : 10.1016/j.crma.2015.10.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.10.022/

Bibliographie
Cité par

[1] M. Bonk; J. Heinonen Smooth quasiregular mappings with branching, Publ. Math. Inst. Hautes Études Sci., Volume 100 (2004), pp. 153-170

[2] K. Fässler; A. Lukyanenko; K. Peltonen Quasiregular mappings on subRiemannian manifolds, J. Geom. Anal. (2015) (forthcoming) | DOI

[3] K. Grove; P. Petersen Bounding homotopy types by geometry, Ann. Math. (2), Volume 128 (1988) no. 1, pp. 195-206

[4] K. Grove; P. Petersen; J.Y. Wu Geometric finiteness theorems via controlled topology, Invent. Math., Volume 99 (1990) no. 1, pp. 205-213

[5] C.Y. Guo Mappings of finite distortion between metric measure spaces, Conform. Geom. Dyn., Volume 19 (2015), pp. 95-121

[6] C.Y. Guo, M. Williams, Porosity of the branch set of discrete open mappings with controlled linear dilatation, preprint, 2015.

[7] C.Y. Guo, S. Nicolussi Golo, M. Williams, Quasiregular mappings between subRimannian manifolds, preprint, 2015.

[8] J. Heinonen The branch set of a quasiregular mapping, Beijing, 2002, Higher Education Press Limited Company, Beijing (2002), pp. 691-700

[9] J. Heinonen; P. Koskela Quasiconformal maps in metric spaces with controlled geometry, Acta Math., Volume 181 (1998), pp. 1-61

[10] J. Heinonen; S. Rickman Quasiregular maps $S^{3} \to S^{3}$ with wild branch sets, Topology, Volume 37 (1998) no. 1, pp. 1-24

[11] J. Heinonen; S. Rickman Geometric branched covers between generalized manifolds, Duke Math. J., Volume 113 (2002) no. 3, pp. 465-529

[12] J. Heinonen; S. Semmes Thirty-three yes or no questions about mappings, measures, and metrics, Conform. Geom. Dyn., Volume 1 (1997), pp. 1-12

[13] J. Heinonen; D. Sullivan On the locally branched Euclidean metric gauge, Duke Math. J., Volume 114 (2002) no. 1, pp. 15-41

[14] O. Martio; S. Rickman; J. Väisälä Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I, Volume 448 (1969), pp. 1-40

[15] O. Martio; S. Rickman; J. Väisälä Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I, Volume 488 (1971), pp. 1-31

[16] L.F. McAuley; E.E. Robinson On Newman's theorem for finite-to-one open mappings on manifolds, Proc. Amer. Math. Soc., Volume 87 (1983) no. 3, pp. 561-566

[17] J. Onninen; K. Rajala Quasiregular mappings to generalized manifolds, J. Anal. Math., Volume 109 (2009), pp. 33-79

[18] P. Petersen A finiteness theorem for metric spaces, J. Differ. Geom., Volume 31 (1990) no. 2, pp. 387-395

[19] S. Rickman Quasiregular Mappings, Ergeb. Math. Grenzgeb. (3), vol. 26, Springer, Berlin, 1993

[20] J. Sarvas The Hausdorff dimension of the branch set of a quasiregular mapping, Ann. Acad. Sci. Fenn. Ser. A I Math., Volume 1 (1975) no. 2, pp. 297-307

[21] S. Semmes Finding curves on general spaces through quantitative topology, with applications for Sobolev and Poincaré inequalities, Sel. Math. New Ser., Volume 2 (1996), pp. 155-295

[22] J. Väisälä Modulus and capacity inequalities for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I, Volume 509 (1972), pp. 1-14

[23] M. Williams, Definition of quasiregularity in metric measure spaces, preprint, 2015.

[24] M. Williams, Bi-Lipschitz embeddability of BLD branched spaces, preprint, 2015.

Cité par Sources :

Commentaires - Politique