  We present a unified description of extremal metrics for the Laplace and
Steklov eigenvalues on manifolds of arbitrary dimension using the notion of
$n$-harmonic maps. Our approach extends the well-known results linking extremal
metrics for eigenvalues on surfaces with minimal immersions and harmonic maps.
In the process, we uncover two previously unknown features of the Steklov
eigenvalues. First, we show that in higher dimensions there is a unique
normalization involving both the volume of the boundary and of the manifold
itself, which leads to meaningful extremal eigenvalue problems. Second, we
observe that the critical points of the eigenvalue functionals in a fixed
conformal class have a natural geometric interpretation provided one considers
the Steklov problem with a density. As an example, we construct a family of
free boundary harmonic annuli in the three-dimensional ball and conjecture that
they correspond to metrics maximizing the first Steklov eigenvalue in their
respective conformal classes.
