Flat chains of finite size in metric spaces
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 1, p. 79-100

In this paper we investigate the notion of flat current in the metric spaces setting, and in particular we provide a definition of size of a flat current with possibly infinite mass. Exploiting the special nature of the 0-dimensional slices and the theory of metric-space valued BV functions we prove that a k-current with finite size T sits on a countably k -rectifiable set, denoted by 𝑠𝑒𝑡(T). Moreover we relate the size measure of T to the geometry of the tangent space Tan (k) (𝑠𝑒𝑡(T),x).

@article{AIHPC_2013__30_1_79_0,
     author = {Ambrosio, Luigi and Ghiraldin, Francesco},
     title = {Flat chains of finite size in metric spaces},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {30},
     number = {1},
     year = {2013},
     pages = {79-100},
     doi = {10.1016/j.anihpc.2012.06.002},
     zbl = {1261.49013},
     mrnumber = {3011292},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2013__30_1_79_0}
}
Ambrosio, Luigi; Ghiraldin, Francesco. Flat chains of finite size in metric spaces. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 1, pp. 79-100. doi : 10.1016/j.anihpc.2012.06.002. http://www.numdam.org/item/AIHPC_2013__30_1_79_0/

[1] F. Almgren, Deformations and multiple-valued functions, Geometric Measure Theory and the Calculus of Variations, Arcata, CA, 1984, Proc. Sympos. Pure Math. vol. 18 (1986), 29-130 | MR 840268 | Zbl 0595.49028

[2] T. Adams, Flat chains in Banach spaces, J. Geom. Anal. 18 (2008), 1-28 | MR 2365666 | Zbl 1148.49037

[3] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr. (2000) | MR 1857292 | Zbl 0957.49001

[4] L. Ambrosio, N. Gigli, G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser (2005) | MR 2129498 | Zbl 1090.35002

[5] L. Ambrosio, Metric space valued functions of bounded variation, Ann. Sc. Norm. Super. 17 (1990), 439-478 | Numdam | MR 1079985 | Zbl 0724.49027

[6] L. Ambrosio, B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), 527-555 | MR 1800768 | Zbl 0966.28002

[7] L. Ambrosio, B. Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), 1-80 | MR 1794185 | Zbl 0984.49025

[8] L. Ambrosio, M. Katz, Flat currents modulo p in metric spaces and filling radius inequalities, Comment. Math. Helv. 86 (2011), 557-592 | MR 2803853 | Zbl 1222.49057

[9] L. Ambrosio, S. Wenger, Rectifiability of flat chains in Banach spaces with coefficients in 𝐙 p , Math. Z. 268 (2011), 477-506 | MR 2805444 | Zbl 1229.49049

[10] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. vol. 580, Springer-Verlag (1977) | MR 467310 | Zbl 0346.46038

[11] E. De Giorgi, Problema di Plateau generale e funzionali geodetici, Atti Semin. Mat. Fis. Univ. Modena 43 (1995), 285-292 | MR 1366062

[12] C. De Lellis, Some fine properties of currents and applications to distributional Jacobians, Proc. Roy. Soc. Edinburgh 132 (2002), 815-842 | MR 1926918 | Zbl 1025.49029

[13] C. De Lellis, E. Spadaro, Q-valued functions revisited, preprint, 2008. | MR 2663735

[14] T. De Pauw, R. Hardt, Size minimization and approximating problems, Calc. Var. Partial Differential Equations 17 (2003), 405-442 | MR 1993962 | Zbl 1022.49026

[15] T. De Pauw, R. Hardt, Rectifiable and flat G-chains in a metric space, Amer. J. Math. 134 (2012), 1-69 | MR 2876138 | Zbl 1252.49070

[16] H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss. vol. 153, Springer-Verlag (1969) | MR 257325 | Zbl 0176.00801

[17] H. Federer, Real flat chains, cochains and variational problems, Indiana Univ. Math. J. 24 (1974/75), 351-407 | MR 348598 | Zbl 0289.49044

[18] H. Federer, W.H. Fleming, Normal and integral current, Ann. of Math. 72 (1960), 458-520 | MR 123260 | Zbl 0187.31301

[19] W.H. Fleming, Flat chains over a finite coefficient group, Trans. Amer. Math. Soc. 121 (1966), 160-186 | MR 185084 | Zbl 0136.03602

[20] F. Ghiraldin, Special functions of bounded higher variation and a functional of Mumford–Shah type in codimension higher than one, in preparation. | Numdam | MR 3182697

[21] B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), 113-123 | MR 1189747 | Zbl 0806.28004

[22] U. Lang, Local currents in metric spaces, J. Geom. Anal. 21 (2011), 683-742 | MR 2810849 | Zbl 1222.49055

[23] F. Morgan, Size-minimizing rectifiable currents, Invent. Math. 96 (1989), 333-348 | MR 989700 | Zbl 0645.49024

[24] M. Petrache, T. Rivière, Weak closure of singular abelian L p -bundles in 3 dimensions, Geom. Funct. Anal. 21 (2011), 1419-1442 | MR 2860193 | Zbl 1242.58009

[25] D. Smets, On some infinite sums of integer valued Diracʼs masses, C. R. Acad. Sci. Paris, Ser. I 334 (2002), 371-374 | MR 1892936 | Zbl 1154.46308

[26] C. Villani, Optimal Transport: Old and New, Grundlehren Math. Wiss. vol. 338, Springer-Verlag, Berlin (2009) | MR 2459454 | Zbl 1156.53003

[27] S. Wenger, Flat convergence for integral currents in metric spaces, Calc. Var. Partial Differential Equations 28 (2007), 139-160 | MR 2284563 | Zbl 1110.53030

[28] B. White, Rectifiability of flat chains, Ann. of Math. 150 (1999), 165-184 | MR 1715323 | Zbl 0965.49024

[29] B. White, The deformation theorem for flat chains, Acta Math. 183 (1999), 255-271 | MR 1738045 | Zbl 0980.49035