  Let $\X = \X[a, b] = \{x: a<|x|<b\}\subset \R^n$ with $0<a<b<\infty$ fixed be
an open annulus and consider the energy functional, \begin{equation*} {\mathbb
F} [u; \X] = \frac{1}{2} \int_\X \frac{|\nabla u|^2}{|u|^2} \, dx,
\end{equation*} over the space of admissible incompressible Sobolev maps
\begin{equation*} {\mathcal A}_\phi(\X) = \bigg\{ u \in W^{1,2}(\X, \R^n) :
\det \nabla u = 1 \text{ {\it a.e.} in $\X$ and $u|_{\partial \X} = \phi$}
\bigg\}, \end{equation*} where $\phi$ is the identity map of $\overline \X$.
Motivated by the earlier works \cite{TA2, TA3} in this paper we examine the
{\it twist} maps as extremisers of ${\mathbb F}$ over ${\mathcal A}_\phi(\X)$
and investigate their minimality properties by invoking the coarea formula and
a symmetrisation argument. In the case $n=2$ where ${\mathcal A}_\phi(\X)$ is a
union of infinitely many disjoint homotopy classes we establish the minimality
of these extremising twists in their respective homotopy classes a result that
then leads to the latter twists being $L^1$-local minimisers of ${\mathbb F}$
in ${\mathcal A}_\phi(\X)$. We discuss variants and extensions to higher
dimensions as well as to related energy functionals.
