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An Allard-type boundary regularity theorem for 2d minimizing currents at smooth curves with arbitrary multiplicity
Publications Mathématiques de l'IHÉS, Volume 140 (2024), pp. 37-154
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We consider integral area-minimizing 2-dimensional currents T in U⊂𝐑 2+n with ∂T=Q[Γ], where Q∈𝐍∖{0} and Γ is sufficiently smooth. We prove that, if q∈Γ is a point where the density of T is strictly below Q+1 2, then the current is regular at q. The regularity is understood in the following sense: there is a neighborhood of q in which T consists of a finite number of regular minimal submanifolds meeting transversally at Γ (and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for Q=1. As a corollary, if Ω⊂𝐑 2+n is a bounded uniformly convex set and Γ⊂∂Ω a smooth 1-dimensional closed submanifold, then any area-minimizing current T with ∂T=Q[Γ] is regular in a neighborhood of Γ.

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DOI: 10.1007/s10240-024-00144-y 
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     author = {Camillo De Lellis and Stefano Nardulli and Simone Steinbr\"uchel},
     title = {An {Allard-type} boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {37--154},
     year = {2024},
     publisher = {Springer International Publishing},
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Camillo De Lellis; Stefano Nardulli; Simone Steinbrüchel. An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity. Publications Mathématiques de l'IHÉS, Volume 140 (2024), pp. 37-154. doi: 10.1007/s10240-024-00144-y

[1.] Some open problems in geometric measure theory and its applications suggested by participants of the 1984 AMS summer institute, in J. E. Brothers (ed.), Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), Proc. Sympos. Pure Math., vol. 44, pp. 441–464, Amer. Math. Soc., Providence, RI, 1986. | MR

[2.] W. K. Allard On boundary regularity for Plateau’s problem, Bull. Am. Math. Soc., Volume 75 (1969), pp. 522-523 | MR | Zbl | DOI

[3.] W. K. Allard On the first variation of a varifold, Ann. Math. (2), Volume 95 (1972), pp. 417-491 | MR | Zbl | DOI

[4.] W. K. Allard On the first variation of a varifold: boundary behavior, Ann. Math. (2), Volume 101 (1975), pp. 418-446 | MR | Zbl | DOI

[5.] F. J. Almgren Almgren’s Big Regularity Paper, 1, World Scientific, River Edge, 2000 | MR | Zbl

[6.] T. Bourni, Allard-type boundary regularity for C 1,α boundaries. ArXiv e-prints, 2010. | MR

[7.] C. De Lellis and E. Spadaro, Q-valued functions revisited, Mem. Amer. Math. Soc., 211(991):vi+79 (2011). | MR

[8.] C. De Lellis; E. Spadaro Regularity of area minimizing currents I: gradient L p estimates, Geom. Funct. Anal., Volume 24 (2014), pp. 1831-1884 | MR | Zbl | DOI

[9.] C. De Lellis; E. Spadaro Multiple valued functions and integral currents, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 14 (2015), pp. 1239-1269 | MR | Zbl

[10.] C. De Lellis; E. Spadaro Regularity of area minimizing currents II: center manifold, Ann. Math. (2), Volume 183 (2016), pp. 499-575 | MR | Zbl | DOI

[11.] C. De Lellis; E. Spadaro Regularity of area minimizing currents III: blow-up, Ann. Math. (2), Volume 183 (2016), pp. 577-617 | MR | Zbl | DOI

[12.] C. De Lellis, E. Spadaro and L. Spolaor, Regularity theory for 2-dimensional almost minimal currents III: blowup. J. Differ. Geom. (2015), ArXiv e-prints, to appear. | MR

[13.] C. De Lellis; E. Spadaro; L. Spolaor Regularity theory for 2-dimensional almost minimal currents II: branched center manifold, Ann. PDE, Volume 3 (2017), p. 18 | MR | Zbl | DOI

[14.] C. De Lellis; E. Spadaro; L. Spolaor Uniqueness of tangent cones for two-dimensional almost-minimizing currents, Commun. Pure Appl. Math., Volume 70 (2017), pp. 1402-1421 | MR | Zbl | DOI

[15.] C. De Lellis, G. De Philippis, J. Hirsch and A. Massaccesi, On the boundary behavior of mass-minimizing integral currents, 2018.

[16.] C. De Lellis; E. Spadaro; L. Spolaor Regularity theory for 2-dimensional almost minimal currents I: Lipschitz approximation, Trans. Am. Math. Soc., Volume 370 (2018), pp. 1783-1801 | MR | Zbl | DOI

[17.] C. De Lellis, S. Nardulli and S. Steinbrüchel, In preparation.

[18.] H. Federer Geometric Measure Theory, 153, Springer, New York, 1969 | MR | Zbl

[19.] R. Hardt; L. Simon Boundary regularity and embedded solutions for the oriented Plateau problem, Ann. Math. (2), Volume 110 (1979), pp. 439-486 | MR | Zbl | DOI

[20.] J. Hirsch Boundary regularity of Dirichlet minimizing Q-valued functions, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 16 (2016), pp. 1353-1407 | MR | Zbl

[21.] J. Hirsch and M. Marini, Uniqueness of tangent cones to boundary points of two-dimensional almost-minimizing currents, 2019, arXiv preprint, | arXiv

[22.] L. Simon Lectures on Geometric Measure Theory, 3, Australian National University Centre for Mathematical Analysis, Canberra, 1983 | MR | Zbl

[23.] B. White Tangent cones to two-dimensional area-minimizing integral currents are unique, Duke Math. J., Volume 50 (1983), pp. 143-160 | MR | Zbl | DOI

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