Spheres under a Cauchy-Riemann inequality in E(κ,τ), and non-CMC Hopf tori
Abstract
For a smooth, connected, oriented surface Σ, topologically a sphere, immersed in the homogeneous space E(κ,τ) with τ≠0, we show that a one-sided Cauchy--Riemann-type inequality on the original Abresch--Rosenberg differential 𝒬AR -- much weaker than requiring 𝒬AR to be holomorphic -- already forces the mean curvature H to be constant. The proof is uniform across every E(κ,τ), 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 𝒬AR 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 π:E(κ,τ)→ M2(κ) 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≡−τ2 -- is a torus.
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BibTeX
@misc{alencar2026spheres,
title = {{Spheres under a Cauchy-Riemann inequality in $E(κ,τ)$, and non-CMC Hopf tori}},
author = {Hilário Alencar and Harold Rosenberg},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23227},
eprint = {2609.23227},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23227},
}
RIS
TY - UNPB
TI - Spheres under a Cauchy-Riemann inequality in E(κ,τ), and non-CMC Hopf tori
AU - Hilário Alencar
AU - Harold Rosenberg
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23227
UR - https://arxiv.org/abs/2609.23227
AB - For a smooth, connected, oriented surface Σ, topologically a sphere, immersed in the homogeneous space E(κ,τ) with τ≠0, we show that a one-sided Cauchy--Riemann-type inequality on the original Abresch--Rosenberg differential 𝒬_AR -- much weaker than requiring 𝒬_AR to be holomorphic -- already forces the mean curvature H to be constant. The proof is uniform across every E(κ,τ), 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 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 π:E(κ,τ)→ M^2(κ) 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≡−τ^2 -- is a torus.
ER -