Critical-point-free energy for fractional-Toledo representations

Let $S_g$ be a closed oriented surface of genus $g\ge 2$. For a reductive representation $\rho :{\pi }_1(S_g)\to \text{PU}(2,1)$, let $E_{\rho }$ be the energy function on Teichmüller space associated to equivariant harmonic maps into ${\text{CH}}^2$. For every positive integer $d$ with $3\nmid d$, all sufficiently large $h$, and every $g>h$, we construct an irreducible reductive representation $${\rho }_{g,h,d}:{\pi }_1(S_g)\to \text{PU}(2,1)$$ with $$\tau ({\rho }_{g,h,d})=2h-2-\frac{2d}{3}\mathrm{\notin }\mathbb{Z},\mathrm{Crit}(E_{{\rho }_{g,h,d}})=\mathrm{\varnothing }\mathrm{.}$$ Consequently, the associated branched-minimal-surface forgetful map is not surjective in these nonintegral Toledo components.