Decomposition of normal currents
Author:
Maciej Zworski
Journal:
Proc. Amer. Math. Soc. 102 (1988), 831839
MSC:
Primary 49F20; Secondary 53C35
MathSciNet review:
934852
Fulltext PDF Free Access
Abstract 
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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.
 [B]
J. B. Brothers (editor), Some open problems in geometric measure theory, Geometric Measure Theory and the Calculus of Variations (W. K. Allard and F. J. Almgren, eds.), Amer. Math. Soc., Providence, R. I., 1986.
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Herbert
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0257325 (41 #1976)
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Wendell
H. Fleming and Raymond
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Arch. Math. (Basel) 11 (1960), 218–222. MR 0114892
(22 #5710)
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Robert
M. Hardt and Jon
T. Pitts, Solving Plateau’s problem for hypersurfaces without
the compactness theorem for integral currents, Geometric measure
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Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1986,
pp. 255–259. MR 840278
(87j:49079), http://dx.doi.org/10.1090/pspum/044/840278
 [H]
R. Harvey, Introduction to geometric measure theory, preprint.
 [M]
Frank
Morgan, On finiteness of the number of stable minimal hypersurfaces
with a fixed boundary, Indiana Univ. Math. J. 35
(1986), no. 4, 779–833. MR 865429
(88b:49059), http://dx.doi.org/10.1512/iumj.1986.35.35042
 [B]
 J. B. Brothers (editor), Some open problems in geometric measure theory, Geometric Measure Theory and the Calculus of Variations (W. K. Allard and F. J. Almgren, eds.), Amer. Math. Soc., Providence, R. I., 1986.
 [F]
 H. Federer, Geometric measure theory, SpringerVerlag, Berlin and New York, 1969. MR 0257325 (41:1976)
 [FR]
 W. H. Fleming and R. Rishel, An integral formula for total gradiant variation, Arch. Math. II (1960), 218222. MR 0114892 (22:5710)
 [HP]
 R. M. Hardt and J. T. Pitts, Solving plateau problem for hypersurfaces without compactness theorem for integral currents, Geometric Measure Theory and the Calculus of Variations, (W. K. Allard and F. J. Almgren, eds.), Amer. Math. Soc., Providence, R. I., 1986. MR 840278 (87j:49079)
 [H]
 R. Harvey, Introduction to geometric measure theory, preprint.
 [M]
 F. Morgan, On finiteness of the number of stable minimal hypersurfaces with a fixed boundary, Indiana Univ. Math. J. 35 (1986), 779834. MR 865429 (88b:49059)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198809348528
PII:
S 00029939(1988)09348528
Article copyright:
© Copyright 1988
American Mathematical Society
