Existence and structure of symmetric Beltrami flows on compact $3$-manifolds
| Autor: | Gerner, Wadim |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | Differential Geometry and its Applications, 78 (2021), 101801 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.difgeo.2021.101801 |
| Popis: | We show that for almost every given symmetry transformation of a Riemannian manifold there exists an eigenvector field of the curl operator, corresponding to a non-zero eigenvalue, which obeys the symmetry. More precisely, given a smooth, compact, oriented Riemannian $3$-manifold $(\bar{M},g)$ with (possibly empty) boundary and a smooth flow of isometries $\phi_t:\bar{M}\rightarrow \bar{M}$ we show that, if $\bar{M}$ has non-empty boundary or if the infinitesimal generator is not purely harmonic, there is a smooth vector field $X$, tangent to the boundary, which is an eigenfield of curl and satisfies $(\phi_t)_{*}X=X$, i.e. is invariant under the pushforward of the symmetry transformation. We then proceed to show that if the quantities involved are real analytic and $(\bar{M},g)$ has non-empty boundary, then Arnold's structure theorem applies to all eigenfields of curl, which obey a symmetry and appropriate boundary conditions. More generally we show that the structure theorem applies to all real analytic vector fields of non-vanishing helicity which obey some nontrivial symmetry. A byproduct of our proof is a characterisation of the flows of real analytic Killing fields on compact, connected, orientable $3$-manifolds with and without boundary. Comment: 23 pages |
| Databáze: | arXiv |
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