Examples of unoriented area-minimizing surfaces

Morgan, Frank

doi:10.1090/S0002-9947-1984-0735418-6

Examples of unoriented area-minimizing surfaces
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by Frank Morgan

Trans. Amer. Math. Soc. 283 (1984), 225-237

DOI: https://doi.org/10.1090/S0002-9947-1984-0735418-6

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Abstract:

A comprehensive study is made of constructions of area-minimizing flat chains modulo two. Many have singularities. For instance, any bounded, area-minimizing submanifold of ${{\mathbf {R}}^n}$ occurs as the singular set of some area-minimizing flat chain modulo two in some ${{\mathbf {R}}^N}$.

References

William K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417–491. MR 307015, DOI 10.2307/1970868
William K. Allard, On the first variation of a varifold: boundary behavior, Ann. of Math. (2) 101 (1975), 418–446. MR 397520, DOI 10.2307/1970934

How to connect minimal surfaces by bridges

3

valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two

Jiri Dadok and Reese Harvey, Calibrations on $\textbf {R}^{6}$, Duke Math. J. 50 (1983), no. 4, 1231–1243. MR 726326, DOI 10.1215/S0012-7094-83-05053-6
Ennio De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43 (Italian). MR 0093649
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Herbert Federer, Real flat chains, cochains and variational problems, Indiana Univ. Math. J. 24 (1974/75), 351–407. MR 348598, DOI 10.1512/iumj.1974.24.24031
Herbert Federer, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 76 (1970), 767–771. MR 260981, DOI 10.1090/S0002-9904-1970-12542-3
Herbert Federer, Some theorems on integral currents, Trans. Amer. Math. Soc. 117 (1965), 43–67. MR 168727, DOI 10.1090/S0002-9947-1965-0168727-0
Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
Wendell H. Fleming, Flat chains over a finite coefficient group, Trans. Amer. Math. Soc. 121 (1966), 160–186. MR 185084, DOI 10.1090/S0002-9947-1966-0185084-5
Wendell H. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2) 11 (1962), 69–90. MR 157263, DOI 10.1007/BF02849427
Robert M. Hardt, On boundary regularity for integral currents or flat chains modulo two minimizing the integral of an elliptic integrand, Comm. Partial Differential Equations 2 (1977), no. 12, 1163–1232. MR 513682, DOI 10.1080/03605307708820058
Reese Harvey and H. Blaine Lawson Jr., Calibrated geometries, Acta Math. 148 (1982), 47–157. MR 666108, DOI 10.1007/BF02392726

The comass ball in

Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
H. Blaine Lawson Jr., The equivariant Plateau problem and interior regularity, Trans. Amer. Math. Soc. 173 (1972), 231–249. MR 308905, DOI 10.1090/S0002-9947-1972-0308905-4
Frank Morgan, Area-minimizing currents bounded by higher multiples of curves, Rend. Circ. Mat. Palermo (2) 33 (1984), no. 1, 37–46. MR 743207, DOI 10.1007/BF02844410

The exterior algebra

and area minimization

Frank Morgan, On the singular structure of three-dimensional, area-minimizing surfaces, Trans. Amer. Math. Soc. 276 (1983), no. 1, 137–143. MR 684498, DOI 10.1090/S0002-9947-1983-0684498-4
Frank Morgan, A smooth curve in $R^{4}$ bounding a continuum of area minimizing surfaces, Duke Math. J. 43 (1976), no. 4, 867–870. MR 415475
Robert Osserman, Minimal varieties, Bull. Amer. Math. Soc. 75 (1969), 1092–1120. MR 276875, DOI 10.1090/S0002-9904-1969-12357-8

The space of

dimensional surfaces that are stationary for a parametric elliptic integrand

Brian White, The structure of minimizing hypersurfaces mod $4$, Invent. Math. 53 (1979), no. 1, 45–58. MR 538683, DOI 10.1007/BF01403190
William P. Ziemer, Integral currents $\textrm {mod}$ $2$, Trans. Amer. Math. Soc. 105 (1962), 496–524. MR 150267, DOI 10.1090/S0002-9947-1962-0150267-3
Herbert Federer, A minimizing property of extremal submanifolds, Arch. Rational Mech. Anal. 59 (1975), no. 3, 207–217. MR 388226, DOI 10.1007/BF00251600