  We investigate the curved thin-film limit of a family of perturbed Dirichlet
energies in the space of $H^1$ Sobolev maps defined in a tubular neighborhood
of an $(n - 1)$-dimensional submanifold $N$ of $\mathbb{R}^n$ and with values
in an $(m - 1)$-dimensional submanifold $M$ of $\mathbb{R}^m$. The perturbation
$\mathsf{K}$ that we consider is represented by a matrix-valued function
defined on $M$ and with values in $\mathbb{R}^{m \times n}$. Under natural
regularity hypotheses on $N$, $M$, and $\mathsf{K}$, we show that the family of
these energies converges, in the sense of $\Gamma$-convergence, to an energy
functional on $N$ of an unexpected form, which is of particular interest in the
theory of magnetic skyrmions. As a byproduct of our results, we get that in the
curved thin-film limit, antisymmetric exchange interactions also manifest under
an anisotropic term whose specific shape depends both on the curvature of the
thin film and the curvature of the target manifold. Various types of
antisymmetric exchange interactions in the variational theory of micromagnetism
are a source of inspiration and motivation for our work.
