On a functional of Kobayashi for Higgs bundles

Abstract

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.