## Quasiconformal mappings and sets of finite perimeter

HTML articles powered by AMS MathViewer

- by James C. Kelly PDF
- Trans. Amer. Math. Soc.
**180**(1973), 367-387 Request permission

## Abstract:

Let $D$ be a domain in ${R^n},n \geqslant 2,f$ a quasiconformal mapping on $D$. We give a definition of*bounding surface of codimension one lying in*$D$, and show that, given a system $\Sigma$ of such surfaces, the image of the restriction of $f$ to “almost every” surface is again a surface. Moreover, on these surfaces, $f$ takes ${H^{n - 1}}$ (Hausdorff $(n - 1)$-dimensional) null sets to ${H^{n - 1}}$ null sets. “Almost every” surface is given a precise meaning via the concept of the module of a system of measures, a generalization of the concept of extremal length.

## References

- Stephen Agard,
*Angles and quasiconformal mappings in space*, J. Analyse Math.**22**(1969), 177–200. MR**252635**, DOI 10.1007/BF02786789
—, - Herbert Federer,
*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR**0257325** - Herbert Federer,
*A note on the Gauss-Green theorem*, Proc. Amer. Math. Soc.**9**(1958), 447–451. MR**95245**, DOI 10.1090/S0002-9939-1958-0095245-2 - Bent Fuglede,
*Extremal length and functional completion*, Acta Math.**98**(1957), 171–219. MR**97720**, DOI 10.1007/BF02404474 - F. W. Gehring,
*Symmetrization of rings in space*, Trans. Amer. Math. Soc.**101**(1961), 499–519. MR**132841**, DOI 10.1090/S0002-9947-1961-0132841-2 - F. W. Gehring,
*Rings and quasiconformal mappings in space*, Trans. Amer. Math. Soc.**103**(1962), 353–393. MR**139735**, DOI 10.1090/S0002-9947-1962-0139735-8 - F. W. Gehring,
*Extremal length definitions for the conformal capacity of rings in space*, Michigan Math. J.**9**(1962), 137–150. MR**140683**, DOI 10.1307/mmj/1028998672 - A. P. Kopylov,
*The behavior of a three-dimensional quasiconformal mapping on plane sections of the region of definition*, Dokl. Akad. Nauk SSSR**167**(1966), 743–746 (Russian). MR**0199382** - H. M. Riemann,
*Über das Verhalten von Flächen unter quasikonformen Abbildungen im Raum*, Ann. Acad. Sci. Fenn. Ser. A. I.**470**(1970), 27 (German). MR**296290** - Ju. G. Rešetnjak,
*Certain geometric properties of functions and mappings with generalized derivatives*, Sibirsk. Mat. .**7**(1966), 886–919 (Russian). MR**0203013** - Jussi Väisälä,
*On quasiconformal mappings in space*, Ann. Acad. Sci. Fenn. Ser. A I No.**298**(1961), 36. MR**0140685** - Jussi Väisälä,
*Two new characterizations for quasiconformality*, Ann. Acad. Sci. Fenn. Ser. A I No.**362**(1965), 12. MR**0174782**
—, - William P. Ziemer,
*Some lower bounds for Lebesgue area*, Pacific J. Math.**19**(1966), 381–390. MR**202971**, DOI 10.2140/pjm.1966.19.381 - William P. Ziemer,
*Extremal length and conformal capacity*, Trans. Amer. Math. Soc.**126**(1967), 460–473. MR**210891**, DOI 10.1090/S0002-9947-1967-0210891-0 - William P. Ziemer,
*Change of variables for absolutely continuous functions*, Duke Math. J.**36**(1969), 171–178. MR**237725**

*Quasiconformal mappings and the moduli of $p$-dimensional surface. families*, Proc. Romanian-Finnish Sem. Teichmüller Spaces Quasiconform. Mappings, Brasov 1969, 1971, pp. 9-48.

*Lectures on $n$-dimensional quasiconformal mappings*, Lecture Notes in Math., vol. 229, Springer-Verlag, Berlin and New York, 1971.

## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**180**(1973), 367-387 - MSC: Primary 30A60; Secondary 28A75
- DOI: https://doi.org/10.1090/S0002-9947-1973-0357783-7
- MathSciNet review: 0357783