Classification of abelian actions with globally hypoelliptic orbitwise laplacian I: The Greenfield-Wallach conjecture on nilmanifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912275797901312 |
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| author | Sandfeldt, Sven |
| author_facet | Sandfeldt, Sven |
| contents | For a $\mathbb{R}^{k}-$action generated by vector fields $X_{1},...,X_{k}$ we define an operator $-(X_{1}^{2}+...+X_{k}^{2})$, the orbitwise laplacian. In this paper, we study and classify $\mathbb{R}^{k}-$actions whose orbitwise laplacian is globally hypoelliptic (GH). In three different settings we prove that any such action is given by a translation action on some compact nilmanifold, (i) when the space is a compact nilmanifold, (ii) when the first Betti number of the manifold is sufficiently large, (iii) when the codimension of the orbitfoliation of the action is $1$. As a consequence, we prove the Greenfield-Wallach conjecture on all nilmanifolds. Along the way, we also calculate the cohomology of GH $\mathbb{R}^{k}-$actions, proving, in particular, that it is always finite dimensional. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15646 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification of abelian actions with globally hypoelliptic orbitwise laplacian I: The Greenfield-Wallach conjecture on nilmanifolds Sandfeldt, Sven Dynamical Systems 37C85 (Primary) 37C15, 37C55 (Secondary) For a $\mathbb{R}^{k}-$action generated by vector fields $X_{1},...,X_{k}$ we define an operator $-(X_{1}^{2}+...+X_{k}^{2})$, the orbitwise laplacian. In this paper, we study and classify $\mathbb{R}^{k}-$actions whose orbitwise laplacian is globally hypoelliptic (GH). In three different settings we prove that any such action is given by a translation action on some compact nilmanifold, (i) when the space is a compact nilmanifold, (ii) when the first Betti number of the manifold is sufficiently large, (iii) when the codimension of the orbitfoliation of the action is $1$. As a consequence, we prove the Greenfield-Wallach conjecture on all nilmanifolds. Along the way, we also calculate the cohomology of GH $\mathbb{R}^{k}-$actions, proving, in particular, that it is always finite dimensional. |
| title | Classification of abelian actions with globally hypoelliptic orbitwise laplacian I: The Greenfield-Wallach conjecture on nilmanifolds |
| topic | Dynamical Systems 37C85 (Primary) 37C15, 37C55 (Secondary) |
| url | https://arxiv.org/abs/2409.15646 |