Slices of maps and Lebesgue area

Ziemer, William P.

doi:10.1090/S0002-9947-1972-0291415-0

Slices of maps and Lebesgue area
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Trans. Amer. Math. Soc. 164 (1972), 139-151 Request permission

Abstract:

For a large class of k dimensional surfaces, S, it is shown that the Lebesgue area of S can be essentially expressed in terms of an integral of the $k - 1$ area of a family, F, of $k - 1$ dimensional surfaces that cover S. The family F is regarded as being composed of the slices of F. The definition of the $k - 1$ area of a surface restricted to one of its slices is formulated in terms of the theory developed by H. Federer, [F3].

References

Lamberto Cesari, Surface area, Annals of Mathematics Studies, No. 35, Princeton University Press, Princeton, N. J., 1956. MR 0074500
Herbert Federer, Some integralgeometric theorems, Trans. Amer. Math. Soc. 77 (1954), 238–261. MR 63686, DOI 10.1090/S0002-9947-1954-0063686-6
Herbert Federer, On Lebesgue area, Ann. of Math. (2) 61 (1955), 289–353. MR 67178, DOI 10.2307/1969917
Hebert Federer, Currents and area, Trans. Amer. Math. Soc. 98 (1961), 204–233. MR 123682, DOI 10.1090/S0002-9947-1961-0123682-0
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
Bent Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171–219. MR 97720, DOI 10.1007/BF02404474
R. E. Fullerton, The Cesari-Cavalieri area of a surface, Duke Math. J. 30 (1963), 151–160. MR 145045, DOI 10.1215/S0012-7094-63-03017-5
Ronald Gariepy, Current valued measures and Geöcze area, Trans. Amer. Math. Soc. 166 (1972), 133–146. MR 293066, DOI 10.1090/S0002-9947-1972-0293066-0
J. H. Michael, The convergence of measures on parametric surfaces, Trans. Amer. Math. Soc. 107 (1963), 140–152. MR 146352, DOI 10.1090/S0002-9947-1963-0146352-3

A class of representations of manifolds

55

56

An analytic characterization of surfaces of finite Lebesgue area

57

58

Tibor Radó, Length and Area, American Mathematical Society Colloquium Publications, Vol. 30, American Mathematical Society, New York, 1948. MR 0024511, DOI 10.1090/coll/030
Raymond Rishel, Area as the integral of lengths of contours, Trans. Amer. Math. Soc. 97 (1960), 95–119. MR 118805, DOI 10.1090/S0002-9947-1960-0118805-2
Paul Slepian, Theory of Lebesgue area of continuous maps of $2$-manifolds into $n$-space, Ann. of Math. (2) 68 (1958), 669–689. MR 98819, DOI 10.2307/1970161
Paul Slepian, On the Lebesgue area of a doubled map, Pacific J. Math. 8 (1958), 613–620. MR 132156, DOI 10.2140/pjm.1958.8.613
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