Spheres under a Cauchy-Riemann inequality in $E(\kappa ,\tau )$, and non-CMC Hopf tori

For a smooth, connected, oriented surface $\mathrm{\Sigma }$, topologically a sphere, immersed in the homogeneous space $E(\kappa ,\tau )$ with $\tau \mathrm{\ne }0$, we show that a one-sided Cauchy–Riemann-type inequality on the original Abresch–Rosenberg differential ${\mathcal{Q}}_{AR}$ – much weaker than requiring ${\mathcal{Q}}_{AR}$ to be holomorphic – already forces the mean curvature $H$ to be constant. The proof is uniform across every $E(\kappa ,\tau )$, including the round sphere, and combines the Bers–Vekua similarity principle with the Poincaré-Hopf index formula for line fields; it is logically independent of the algebraic argument used, in the companion paper~\cite{AlencarRosenberg2026}, to characterize CMC immersions by holomorphy of ${\mathcal{Q}}_{AR}$ alone. We also show that the topological hypothesis is sharp: on every Berger sphere, including the round one, there exist compact non-CMC tori – the preimages under the Hopf-type submersion $\pi :E(\kappa ,\tau )\to M^2(\kappa )$ of simple closed curves in the base with nonconstant geodesic curvature – satisfying the same inequality with a constant bound. As an elementary consequence of the classical Poincaré-Hopf theorem for line fields, we record that every closed surface on which $\det S<0$ everywhere – a condition satisfied, in particular, by every Hopf tube, where $\det S\equiv -{\tau }^2$ – is a torus.