An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity

De Lellis, Camillo; Nardulli, Stefano; Steinbrüchel, Simone

doi:10.1007/s10240-024-00144-y

Article

An Allard-type boundary regularity theorem for

2 d

minimizing currents at smooth curves with arbitrary multiplicity

De Lellis, Camillo ; Nardulli, Stefano ; Steinbrüchel, Simone

Publications Mathématiques de l'IHÉS, Tome 140 (2024), pp. 37-154

Résumé

We consider integral area-minimizing 2-dimensional currents $T$ in $U \subset 𝐑^{2 + n}$ with $\partial T = Q [[Γ]]$ , where $Q \in 𝐍 ∖ {0}$ and $Γ$ is sufficiently smooth. We prove that, if $q \in Γ$ is a point where the density of $T$ is strictly below $\frac{Q + 1}{2}$ , 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 $Ω \subset 𝐑^{2 + n}$ is a bounded uniformly convex set and $Γ \subset \partial Ω$ a smooth 1-dimensional closed submanifold, then any area-minimizing current $T$ with $\partial T = Q [[Γ]]$ is regular in a neighborhood of $Γ$ .

Publié le : 2024-12-28

MR Zbl

DOI : 10.1007/s10240-024-00144-y

@article{PMIHES_2024__140__37_0,
     author = {De Lellis, Camillo and Nardulli, Stefano and Steinbr\"uchel, Simone},
     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},
     address = {Cham},
     volume = {140},
     doi = {10.1007/s10240-024-00144-y},
     mrnumber = {4824747},
     zbl = {1555.49063},
     language = {en},
     url = {https://www.numdam.org/articles/10.1007/s10240-024-00144-y/}
}

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De Lellis, Camillo; Nardulli, Stefano; Steinbrüchel, Simone. An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity. Publications Mathématiques de l'IHÉS, Tome 140 (2024), pp. 37-154. doi: 10.1007/s10240-024-00144-y

Bibliographie
Cité par

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

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

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

[5.] Almgren, F. J. Jr. 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.] De Lellis, C.; Spadaro, E. Regularity of area minimizing currents I: gradient $L^{p}$ estimates, Geom. Funct. Anal., Volume 24 (2014), pp. 1831-1884 | MR | Zbl | DOI

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

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

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

[14.] De Lellis, C.; Spadaro, E.; Spolaor, L. 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.] De Lellis, C.; Spadaro, E.; Spolaor, L. 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.] Federer, H. Geometric Measure Theory, 153, Springer, New York, 1969 | MR | Zbl

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

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

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

Cité par Sources :