Abstract
We consider integral area-minimizing 2-dimensional currents \(T\) in \(U\subset \mathbf {R}^{2+n}\) with \(\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]\), where \(Q\in \mathbf {N} \setminus \{0\}\) and \(\Gamma \) is sufficiently smooth. We prove that, if \(q\in \Gamma \) 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 \(\Gamma \) (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 \(\Omega \subset \mathbf {R}^{2+n}\) is a bounded uniformly convex set and \(\Gamma \subset \partial \Omega \) a smooth 1-dimensional closed submanifold, then any area-minimizing current \(T\) with \(\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]\) is regular in a neighborhood of \(\Gamma \).
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Acknowledgements
C.D.L. acknowledges support from the National Science Foundation through the grant FRG-1854147. The second author was partially sponsored by “Bolsa no Exterior-Pesquisa” (Fapesp BPE, 2018/22938-4), “Jovens Pesquisadores em Centros Emergentes” (JP-FAPESP, 21/05256-0), and by “Bolsa de Produtividade em Pesquisa 2” (CNPq, 302717/2017-0), Brazil. Moreover, the second author would like to thank the Department of Mathematics of Princeton University for the year spent at that institution as a visiting fellow, during which large parts of the manuscript were written, and the Institute for Advanced Study of Princeton for the hospitality offered during various short term visits. All three authors wish to thank the anonymous referee who suggested various improvements to the manuscript.
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De Lellis, C., Nardulli, S. & Steinbrüchel, S. An Allard-type boundary regularity theorem for \(2d\) minimizing currents at smooth curves with arbitrary multiplicity. Publ.math.IHES (2024). https://doi.org/10.1007/s10240-024-00144-y
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DOI: https://doi.org/10.1007/s10240-024-00144-y