  In this paper, we derive a maximum principle for a type of elliptic systems
and apply it to analyze the Hitchin equation for cyclic Higgs bundles. We show
several domination results on the pullback metric of the (possibly branched)
minimal immersion $f$ associated to cyclic Higgs bundles. Also, we obtain a
lower and upper bound of the extrinsic curvature of the image of $f$. As an
application, we give a complete picture for maximal
$Sp(4,\mathbb{R})$-representations in the $2g-3$ Gothen components and the
Hitchin components.
