Quasiregular mappings in even dimensions
In their introduction to this long important paper the authors say that the paper has grown from their study of a paper of \textit{S. K. Donaldson} and \textit{D. P. Sullivan} [DS] [Acta Math. 163, No. 3/4, 181-252 (1989; Zbl 0704.57008)]. Using and developing the ideas of [DS] the authors study differential forms with integrable coefficients and Hodge theory on \(L^ p\)-spaces and with great success apply these methods to investigate several topics and open problems in the theory of quasiconformal and quasiregular maps in \(R^ n\), \(n \geq 2\). Among such topics are (a) a definition of a new Beurling-Ahlfors type singular integral operator \(S\) suited for higher dimensional studies, (b) regularity questions in Liouville's theorem in even dimensions, (c) discovery of new differential equations satisfied by quasiregular maps in even dimensions, (d) new definition of quasiregular maps, (e) derivation of Caccioppoli type estimates. The authors give various examples showing the sharpness of their results and also present various conjectures. It is not possible to adequately describe even the main results of the paper in this review. Therefore we refer the reader to a recent survey of the first author [in Lect. Notes Math. 1508, 39-64 (1992; reviewed below)] where also this paper is discussed and conclude by what the authors regard as their most important result. This result deals with the study of exceptional sets of bounded quasiregular maps. Here the main problem is the following one. Let \(E\) be a compact subset of the unit ball \(B^ n\) of measure zero and let \(f:B^ n \backslash E \to R^ n\) be a bounded quasiregular map. Under which conditions on \(E\) and \(f\) the map \(f\) has an extension to \(E\)? \textit{Yu. G. Reshetnyak} [Sib. Mat. Zh. 10, 1300-1310 (1969; Zbl 0201.098)] and \textit{O. Martio}, \textit{S. Rickman} and \textit{J. Väisälä} [Ann. Acad. Sci. Fenn., Ser. A I 465, 1-13 (1970; Zbl 0197.057)] settled the case when \(E\) is of zero \(n\)-capacity. But what about the case when \(E\) is of positive Hausdorff dimension? With little success this problem was studied by \textit{E. A. Poletskij} [Mat. Sb., Nov. Ser. 92(134), 242-256 (1973; Zbl 0281.30017)] and the reviewer [Ann. Acad. Sci. Fenn., Ser. A I, Diss. 11, 44 p. (1976; Zbl 0362.30024)]. The authors obtain now a sufficient condition for the removability in even dimensions \(n=2l\) in terms of a condition that involves \(lp\)-capacity of \(E\), \(p\)-norm of the \(S\)-operator and the ``Beltrami coefficient'' (which the authors define) of \(f\). In this context, related simultaneous results should be mentioned, namely the papers of \textit{P. Koskela} and \textit{O. Martio} [Ann. Acad. Sci. Fenn., Ser. AI 15, No. 2, 381-399 (1990; Zbl 0717.30015)], of \textit{P. Järvi} and the reviewer [J. Reine Angew. Math. 424, 31-45 (1992; Zbl 0733.30017)] and of S. Rickman (to appear)]. The first author has later extended some of the results also to the case of odd dimensions [Ann. Math. 136, 651-685 (1992; reviewed below)].
scientific article; zbMATH DE number 205188
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