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<p>We construct an explicit example of a smooth isotopy {<i>ξ</i><sub><i>t</i></sub>}<sub><i>t</i>∈[0,1]</sub> of volume- and orientation-preserving diffeomorphisms on [0, 1]<sup><i>n</i></sup> (<i>n</i> ≥ 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 (<i>M, g</i>), a “topologically complicated” Riemannian manifold-with-boundary. However, there exists another smooth isotopy that coincides with {<i>ξ</i><sub><i>t</i></sub>} at <i>t</i> = 0 and <i>t</i> = 1 but of finite total kinetic energy.</p>

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A Smooth Isotopy of Volume-preserving Diffeomorphisms on Unit Cube Saving Energy through Extra Dimensions

  • 期刊论文

  • 发表日期：

    2025年11月15日

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  • Siran Li

摘要

We construct an explicit example of a smooth isotopy {ξt}t∈[0,1] of volume- and orientation-preserving diffeomorphisms on [0, 1]n (n ≥ 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 {ξt} at t = 0 and t = 1 but of finite total kinetic energy.