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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2409.00958 |
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| _version_ | 1866909302488301568 |
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| author | Cheng, Wei Wei, Wenxue |
| author_facet | Cheng, Wei Wei, Wenxue |
| contents | Let $(M,g)$ be a closed, connected and orientable Riemannian manifold with nonnegative Ricci curvature. Consider a Lagrangian $L(x,v):TM\to\R$ defined by $L(x,v):=\frac 12g_x(v,v)-ω(v)+c$, where $c\in\R$ and $ω$ is a closed 1-form. From the perspective of differential geometry, we estimate the Laplacian of the weak KAM solution $u$ to the associated Hamilton-Jacobi equation $H(x,du)=c[L]$ in the barrier sense. This analysis enables us to prove that each weak KAM solution $u$ is constant if and only if $ω$ is a harmonic 1-form. Furthermore, we explore several applications to the Mather quotient and Mañé's Lagrangian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00958 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A geometric approach to Mather quotient problem Cheng, Wei Wei, Wenxue Dynamical Systems Differential Geometry 35F21, 49L25, 37J50 Let $(M,g)$ be a closed, connected and orientable Riemannian manifold with nonnegative Ricci curvature. Consider a Lagrangian $L(x,v):TM\to\R$ defined by $L(x,v):=\frac 12g_x(v,v)-ω(v)+c$, where $c\in\R$ and $ω$ is a closed 1-form. From the perspective of differential geometry, we estimate the Laplacian of the weak KAM solution $u$ to the associated Hamilton-Jacobi equation $H(x,du)=c[L]$ in the barrier sense. This analysis enables us to prove that each weak KAM solution $u$ is constant if and only if $ω$ is a harmonic 1-form. Furthermore, we explore several applications to the Mather quotient and Mañé's Lagrangian. |
| title | A geometric approach to Mather quotient problem |
| topic | Dynamical Systems Differential Geometry 35F21, 49L25, 37J50 |
| url | https://arxiv.org/abs/2409.00958 |