A smooth isotopy of volume-preserving diffeomorphisms on unit cube saving energy through extra dimensions
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866908794586398720 |
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| author | Li, Siran |
| author_facet | Li, Siran |
| contents | We construct an explicit example of a smooth isotopy $\{ξ_t\}_{t \in [0,1]}$ of volume- and orientation-preserving diffeomorphisms on $[0,1]^n$ ($n \geq 3$) that has infinite total kinetic energy. This isotopy has no self-cancellation and is supported on countably many disjoint tubular neighbourhoods of homothetic copies of the isometrically embedded image of $(M,g)$, a "topologically complicated" Riemannian manifold-with-boundary. However, there exists another smooth isotopy that coincides with $\{ξ_t\}$ at $t=0$ and $t=1$ but of finite total kinetic energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07057 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A smooth isotopy of volume-preserving diffeomorphisms on unit cube saving energy through extra dimensions Li, Siran Differential Geometry Analysis of PDEs We construct an explicit example of a smooth isotopy $\{ξ_t\}_{t \in [0,1]}$ of volume- and orientation-preserving diffeomorphisms on $[0,1]^n$ ($n \geq 3$) that has infinite total kinetic energy. This isotopy has no self-cancellation and is supported on countably many disjoint tubular neighbourhoods of homothetic copies of the isometrically embedded image of $(M,g)$, a "topologically complicated" Riemannian manifold-with-boundary. However, there exists another smooth isotopy that coincides with $\{ξ_t\}$ at $t=0$ and $t=1$ but of finite total kinetic energy. |
| title | A smooth isotopy of volume-preserving diffeomorphisms on unit cube saving energy through extra dimensions |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2309.07057 |