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arXiv · 2309.07057

A smooth isotopy of volume-preserving diffeomorphisms on unit cube saving energy through extra dimensions

Siran Li
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Abstract

We construct an explicit example of a smooth isotopy $\{\xi_t\}_{t \in [0,1]}$ 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 $\{\xi_t\}$ at $t=0$ and $t=1$ but of finite total kinetic energy.

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Siran Li. 2023-09-13. A smooth isotopy of volume-preserving diffeomorphisms on unit cube saving energy through extra dimensions. https://arxiv.org/abs/2309.07057

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