Critical-point-free energy for fractional-Toledo representations

Abstract

Let $S_g$ be a closed oriented surface of genus $g\ge2$. For a reductive representation $\rho:\pi_1(S_g)\to\PU(2,1)$, let $E_\rho$ be the energy function on Teichm\"uller space associated to equivariant harmonic maps into $\CH^2$. For every positive integer $d$ with $3\nmid d$, all sufficiently large $h$, and every $g>h$, we construct an irreducible reductive representation \[ \rho_{g,h,d}:\pi_1(S_g)\to\PU(2,1) \] with \[ \tau(\rho_{g,h,d})=2h-2-\frac{2d}{3}\notin\mathbb Z, \qquad \operatorname{Crit}(E_{\rho_{g,h,d}})=\varnothing. \] Consequently, the associated branched-minimal-surface forgetful map is not surjective in these nonintegral Toledo components.