An $\infty$-Laplacian for differential forms, and calibrated laminations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913296683106304 |
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| author | Backus, Aidan |
| author_facet | Backus, Aidan |
| contents | Motivated by Thurston and Daskalopoulos--Uhlenbeck's approach to Teichmüller theory, we study the behavior of $q$-harmonic functions and their $p$-harmonic conjugates in the limit as $q \to 1$, where $1/p + 1/q = 1$. The $1$-Laplacian is already known to give rise to laminations by minimal hypersurfaces; we show that the limiting $p$-harmonic conjugates converge to calibrations $F$ of the laminations. Moreover, we show that the laminations which are calibrated by $F$ are exactly those which arise from the $1$-Laplacian. We also explore the limiting dual problem as a model problem for the optimal Lipschitz extension problem, which exhibits behavior rather unlike the scalar $\infty$-Laplacian. In a companion work, we will apply the main result of this paper to associate to each class in $H^{d - 1}$ a lamination in a canonical way, and study the duality of the stable norm on $H_{d - 1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02215 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An $\infty$-Laplacian for differential forms, and calibrated laminations Backus, Aidan Analysis of PDEs Differential Geometry primary: 35J94, secondary: 35J92, 53C38, 49N15, 49Q20 Motivated by Thurston and Daskalopoulos--Uhlenbeck's approach to Teichmüller theory, we study the behavior of $q$-harmonic functions and their $p$-harmonic conjugates in the limit as $q \to 1$, where $1/p + 1/q = 1$. The $1$-Laplacian is already known to give rise to laminations by minimal hypersurfaces; we show that the limiting $p$-harmonic conjugates converge to calibrations $F$ of the laminations. Moreover, we show that the laminations which are calibrated by $F$ are exactly those which arise from the $1$-Laplacian. We also explore the limiting dual problem as a model problem for the optimal Lipschitz extension problem, which exhibits behavior rather unlike the scalar $\infty$-Laplacian. In a companion work, we will apply the main result of this paper to associate to each class in $H^{d - 1}$ a lamination in a canonical way, and study the duality of the stable norm on $H_{d - 1}$. |
| title | An $\infty$-Laplacian for differential forms, and calibrated laminations |
| topic | Analysis of PDEs Differential Geometry primary: 35J94, secondary: 35J92, 53C38, 49N15, 49Q20 |
| url | https://arxiv.org/abs/2404.02215 |