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

TL;DR

The paper constructs a smooth volume-preserving isotopy on a high-dimensional cube with infinite energy, yet demonstrates an alternative isotopy with the same endpoints and finite energy, highlighting energy-saving via extra dimensions.

Contribution

It provides an explicit example of an isotopy with infinite energy and shows how an alternative isotopy can reduce energy, revealing new insights into volume-preserving diffeomorphisms.

Findings

Explicit isotopy with infinite kinetic energy

Existence of finite-energy isotopy with same endpoints

Energy can be saved using extra dimensions

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.