  We define a functional ${\cal J}(h)$ for the space of Hermitian metrics on an
arbitrary Higgs bundle over a compact K\"ahler manifold, as a natural
generalization of the mean curvature energy functional of Kobayashi for
holomorphic vector bundles \cite{Kobayashi}, and study some of its basic
properties. We show that ${\cal J}(h)$ is bounded from below by a nonnegative
constant depending on invariants of the Higgs bundle and the K\"ahler manifold,
and that when achieved, its absolute minima are Hermite-Yang-Mills metrics. We
derive a formula relating ${\cal J}(h)$ and another functional ${\cal I}(h)$,
closely related to the Yang-Mills-Higgs functional \cite{Bradlow-Wilkin,
Wentworth}, which can be thought of as an extension of a formula of Kobayashi
for holomorphic vector bundles to the Higgs bundles setting. Finally, using
1-parameter families in the space of Hermitian metrics on a Higgs bundle, we
compute the first variation of ${\cal J}(h)$, which is expressed as a certain
$L^{2}$-Hermitian inner product. It follows that a Hermitian metric on a Higgs
bundle is a critical point of ${\cal J}(h)$ if and only if the corresponding
Hitchin--Simpson mean curvature is parallel with respect to the
Hitchin--Simpson connection.
