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

We construct an explicit example of a smooth isotopy $\{{\xi }_t{\}}_{t\in [0,1]}$ of volume- and orientation-preserving diffeomorphisms on $[0,1{]}^n$ ($n\ge 3$) 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.