×

zbMATH — the first resource for mathematics

Estimates for the energy integral of quasiregular mappings on Riemannian manifolds and isoperimetry. (English) Zbl 1079.30508
Summary: The rate of growth of the energy integral of a quasiregular mapping \(f\:\mathcal X\to \mathcal Y\) is estimated in terms of a special isoperimetric condition on \(\mathcal Y\). The estimate leads to new Phragmén-Lindelöf type theorems.
MSC:
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] L. Ahlfors: Zur Theorie Überlagerungsflächen. Acta Math. 65 (1935), 157-194. · Zbl 0012.17204
[2] V. A. Botvinnik: Phragmén-Lindelöf’s theorems for space mappings with boundary distortion. Dissertation, Volgograd (1983), 1-96.
[3] Yu. D. Burago and V. A. Zalgaller: Geometric Inequalities. Nauka, Moscow, 1980. · Zbl 0436.52009
[4] C. Croke: Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. École Norm. Sup. (4) Ser. 13 (1980), 419-435. · Zbl 0465.53032
[5] H. Federer: Geometric Measure Theory. Springer-Verlag, Berlin-Heidelberg-New York, 1969. · Zbl 0176.00801
[6] S. Granlund, P. Lindqvist, and O. Martio: Phragmén-Lindelöf’s and Lindelöf’s theorem. Ark. Mat. 23 (1985), 103-128. · Zbl 0594.30022
[7] M. Gromov: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), 307-347. · Zbl 0592.53025
[8] J. Heinonen, T. Kilpeläinen and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. Clarendon Press, 1993. · Zbl 0780.31001
[9] D. Hoffman and J. Spruck: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure Appl. Math. 27 (1974), 715-727. · Zbl 0295.53025
[10] I. Holopainen and S. Rickman: Classification of Riemannian manifolds in nonlinear potential theory. Potential Analysis 2 (1993), 37-66. · Zbl 0771.53019
[11] A. S. Kronrod: On functions of two variables. Uspekhi Mat. Nauk 5 (1950), 24-134.
[12] O. Martio, V. Miklyukov and M. Vuorinen: Differential forms and quasiregular mappings on Riemannian manifolds. XVIth Rolf Nevanlinna Colloquium (I. Laine and O. Martio, Walter de Gruyter &Co, 1996, pp. 151-159. · Zbl 0899.58065
[13] O. Martio, V. Miklyukov and M. Vuorinen: Phragmén - Lindelöf’s principle for quasiregular mappings and isoperimetry. Dokl. Akad. Nauk 347 (1996), 303-305. · Zbl 0886.30020
[14] V. M. Miklyukov: Asymptotic properties of subsolutions of quasilinear equations of elliptic type and mappings with bounded distortion. Mat. Sb. 11 (1980), 42-66.
[15] P. Pansu: Quasiconformal mappings and manifolds of negative curvature. Curvature and Topology of Riemannian Manifolds. Proceed. 17th Int. Taniguchi Symp., Katata, Japan, Aug. 26-31, 1985. · Zbl 0592.53031
[16] E. Phragmén and E. Lindelöf: Sur une extension d’un principe classique de l’analyse et sur quelques propriétés des fonctions monogenènes dans le voisinage d’un point singulier. Acta Math. 31 (1908), 381-406. · JFM 39.0465.01
[17] S. Rickman and M. Vuorinen: On the order of quasiregular mappings. Ann. Acad. Sci. Fenn. Math. 7 (1982), 221-231. · Zbl 0516.30018
[18] M. Vuorinen: Conformal Geometry and Quasiregular Mappings. Lecture Notes in Math., 1319, Springer-Verlag. · Zbl 0646.30025
[19] S. T. Yau: Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold. Ann. Sci. École Norm. Sup. 8 (1975), 487-507. · Zbl 0325.53039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.