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

Abstract

For a smooth, connected, oriented surface $\Sigma$, topologically a sphere, immersed in the homogeneous space $E(\kappa,\tau)$ with $\tau\ne0$, 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 Poincar\'e-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 Poincar\'e-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.