Existence and partial regularity of Legendrian area-minimizing currents
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909223906967552 |
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| author | Orriols, Gerard |
| author_facet | Orriols, Gerard |
| contents | We show that Legendrian integral currents in a contact manifold that locally minimize the mass among Legendrian competitors have a regular set which is open and dense in their support. We apply this to show existence and partial regularity of solutions of the Legendrian Plateau problem in the $n$th Heisenberg group for an arbitrary horizontal $(n-1)$-cycle as prescribed boundary, and of mass-minimizing Legendrian integral currents in any $n$-dimensional homology class of a closed contact $(2n+1)$-manifold. In the case of the Heisenberg group, our result applies to Ambrosio--Kirchheim metric currents with respect to the Carnot--Carathéodory distance. Our results do not assume any compatibility between the subriemannian metric and the symplectic form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09378 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and partial regularity of Legendrian area-minimizing currents Orriols, Gerard Differential Geometry Analysis of PDEs Metric Geometry We show that Legendrian integral currents in a contact manifold that locally minimize the mass among Legendrian competitors have a regular set which is open and dense in their support. We apply this to show existence and partial regularity of solutions of the Legendrian Plateau problem in the $n$th Heisenberg group for an arbitrary horizontal $(n-1)$-cycle as prescribed boundary, and of mass-minimizing Legendrian integral currents in any $n$-dimensional homology class of a closed contact $(2n+1)$-manifold. In the case of the Heisenberg group, our result applies to Ambrosio--Kirchheim metric currents with respect to the Carnot--Carathéodory distance. Our results do not assume any compatibility between the subriemannian metric and the symplectic form. |
| title | Existence and partial regularity of Legendrian area-minimizing currents |
| topic | Differential Geometry Analysis of PDEs Metric Geometry |
| url | https://arxiv.org/abs/2406.09378 |