PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 102, Number 4, April 1988 DECOMPOSITION OF NORMAL CURRENTS MACIEJ ZWORSKI (Communicated by David G. Ebin) ABSTRACT. As an answer to a question of Frank Morgan, it is shown that there exist normal currents which cannot be represented as convex integrals of rectifiable currents. However, under certain additional hypotheses, such decompositions exist. Examples are given to indicate that such hypotheses are necessary. 1. Introduction. JThe purpose of this note is to answer a question (*) posed by Frank Morgan (compare Problem 3.8 in [B]): Does every normal current in codimension one admit a nice mass decomposition as an integral of rectifiable currents? Specifically for m = n — 1 and N E NTO(Rn) does there exist a family {R^juen such that (1) i2weRm(R") and O is some measure space, (2) N = [ Ru dto, Jn (3) \\N\\= f \\R„\\du,, Ja (4) \\dN\\ = [ WdRuWdcj, Jn Condition (4) means that the mass decomposition is simultaneously a boundary mass decomposition. This problem has been first studied by Fleming and Rishel [FR] (see also [F, 4.9.15(13)]), who proved existence of a decomposition satisfying (l)-(4) for currents N E Nn_, (R") with dN = 0. In this case N = d(En |_/) and (5) N = Jd(Enl{x:f(x)>s})dLl, (6) \\N\\=J\\d(Eni{x:f(x)>s})\\dLl. As remarked by Morgan this also implies decomposition in the case of extremal boundary. Recently Hardt and Pitts [HP] solved Plateau's Problem for hypersurfaces us- ing an ingeneous modification of the level set argument (5) from [FR]. Their proof shows the existence of a decomposition satisfying (l)-(4) in the case of N E Nn_,(Rn) with ¿57V E Rn_2(Rn) (Theorem 1). Received by the editors January 3, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 49F20; Secondary 53C12. 'The notation throughout the paper is taken from [F], ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page 831 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use