Proceedings of the American Mathematical Society

Author(s): O. Martio; V. M. Miklyukov; M. Vuorinen
Journal: Proc. Amer. Math. Soc. 135 (2007), 2433-2442.
MSC (2000): Primary 30C62, 30C65
Posted: April 5, 2007
Retrieve article in: PDF DVI PostScript

Abstract: A variational inequality for the images of

-dimensional hyperplanes under quasiconformal maps of the

-dimensional Euclidean space is proved when $1\le k\le n-2.$

1.: L. AHLFORS, Quasiconformal reflections, Acta Math. 109 (1963), 291-301.
2.: G.D. ANDERSON, M.K. VAMANAMURTHY, AND M.K. VUORINEN, Conformal invariants, inequalities, and quasiconformal maps, Canadian Math. Soc., Ser. of Monographs and Advanced texts, A Wiley - Interscience Publication, John Wiley&Sons, Inc., New-York - Chichester - Weinheim - Brisbane - Singapore - Toronto, 1997.
3.: C. BANDLE, Isoperimetric inequalities and applications, Pitman Advanced Publishing, Boston-London-Melbourne, 1980.
4.: E.F. BECKENBACH AND R. BELLMAN, Inequalities, Springer-Verlag, Berlin - Göttingen - Heidelberg, 1961.
5.: M. BONK AND J. HEINONEN, Smooth quasiregular mappings with branching, Publ. Math. Inst. Hautes Études Sci. No. 100 (2004), 153-170.
6.: L.C. EVANS AND R.F. GARIEPY, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC PRESS, Boca Raton - New York - London - Tokyo, 1992.
7.: D. FRANKE, O. MARTIO, V.M. MIKLYUKOV, M. VUORINEN, AND R. WISK, Quasiregular mappings and ${\mathcal WT}$ -classes of differential forms on Riemannian manifolds, Pacific J. Math. 202 (2002), 73-92.
8.: F.W. GEHRING, Uniform domains and the ubiquitous quasidisk, Jahresber. Deutsch. Math.-Verein, 89 (1987), 88-103.
9.: M. GROMOV, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347.
10.: J. HEINONEN, T. KILPELÄINEN AND O. MARTIO, Nonlinear potential theory of degenerate elliptic equations, Clarendon Press, 1993.
11.: S.L. KRUSHKAL, Quasiconformal mirrors, Sibirsk. Mat. Zh. 40 (1999), 880-892.
12.: O. MARTIO, V. MIKLYUKOV, S. PONNUSAMY, AND M. VUORINEN, On some properties of quasiplanes, Results Math. 42 (2002), 107-113.
13.: P. MATTILA AND M.VUORINEN, Linear approximation property, Minkowski dimension and quasiconformal spheres, J. London Math. Soc. (2) 42 (1990), 249-266.
14.: V.M. MIKLYUKOV, On quasiconformally flat surfaces in Riemannian manifolds, Izv. RAN (ser. math.), 67 (2003), 83-106.
15.: J. SIMONS, Minimal varieties in Riemannian manifolds, Ann. Math., 88 (1968), No. 1, 62-105.
16.: J. VÄISÄLÄ, M. VUORINEN AND H. WALLIN, Thick sets and quasisymmetric maps, Nagoya Math. J. 135 (1994), 121-148.
17.: M. VUORINEN, Geometric properties of quasiconformal maps and special functions. I. Quasiconformal maps and spheres. (English. English, Polish summary) Bull. Soc. Sci. Lett. $\L$ ódz Sér. Rech. Déform. 24 (1997), 7-22. (see also Errata: "Geometric properties of quasiconformal maps and special functions. I, II, III [Bull. Soc. Sci. Lett. $\L$ ódz Sér. Rech. Déform. 24 (1997), 7-22; ibid., 23-35; ibid., 37-58; ]. Bull. Soc. Sci. Lett. $\L$ ódz Sér. Rech. Déform. 26 (1998).
18.: M. VUORINEN, O. MARTIO, AND V.M. MIKLYUKOV, On geometric structure of quasiplanes, Proc. of Dept. of Mathematical Analysis and Function Theory, Volgograd State Univ., 2002, 21-31, ISBN 5-85534-531-9.
19.: I. PRAUSE, Flatness properties of quasispheres, Preprint, University of Helsinki 2007.
20.: I. PRAUSE, Distortion of dimension under quasiconformal mappings, Ph.D. thesis, University of Helsinki 2007.

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 30C62, 30C65

O. Martio
Affiliation: Department of Mathematics and Statistics, University of Helsinki, 00014 Helsinki, Finland
Email: martio@cc.helsinki.fi

V. M. Miklyukov
Affiliation: Mathematics Department, Volgograd State University, 2 Prodolnaya 30, Volgograd 400062, Russia
Email: miklyuk@hotmail.com

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS