Only constant mean curvature surfaces in E(κ,τ), τ≠0, have a holomorphic original Abresch-Rosenberg differential
Abstract
For immersed surfaces in the homogeneous spaces E(κ,τ) with τ≠0, we prove that the original Abresch--Rosenberg differential [2(H+iτ)p−(κ−4τ2)A2] dζ2 is holomorphic if and only if the mean curvature H is constant; here p dζ2 is the Hopf differential and A=⟨ξ,∂ζ⟩. The result is local: no completeness, compactness or analyticity is assumed, and there are no exceptional surfaces. Where dH≠0, holomorphy forces the shape operator into the normal form (0&τ τ&2H) in the frame (∇ H/|∇ H|,J∇ H/|∇ H|). The round case κ=4τ2 is settled by an elementary argument; for κ≠4τ2 the structure equations become a rational system whose solutions are algebraic, and an embedding into a field of Puiseux series excludes every branch. In M2(κ)×ℝ, κ≠0, where the statement fails, we show that a non-CMC surface with holomorphic original differential has nowhere vanishing dH and constant angle function, and belongs to an explicit family; in particular, closed surfaces with holomorphic original differential are CMC there. We also show that the Hopf-cylinder alternative in a theorem of Dorfmeister, Inoguchi and Kobayashi contains no non-CMC surface.
Related
- Constant Chern holomorphic sectional curvature on two-step solvmanifolds
- Hermitian manifolds with constant holomorphic sectional curvature or real bisectional curvature
- Hermitian connections with parallel torsion and constant holomorphic sectional curvature
- Biholomorphic rigidity and symmetry breaking for domains with Bergman metric of constant positive holomorphic sectional curvature
- Hermitian threefolds with constant holomorphic sectional curvature
BibTeX
@misc{alencar2026only,
title = {{Only constant mean curvature surfaces in $E(κ,τ)$, $τ\ne0$, have a holomorphic original Abresch-Rosenberg differential}},
author = {Hilário Alencar and Harold Rosenberg},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41197},
eprint = {2610.41197},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41197},
}
RIS
TY - UNPB
TI - Only constant mean curvature surfaces in E(κ,τ), τ≠0, have a holomorphic original Abresch-Rosenberg differential
AU - Hilário Alencar
AU - Harold Rosenberg
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41197
UR - https://arxiv.org/abs/2610.41197
AB - For immersed surfaces in the homogeneous spaces E(κ,τ) with τ≠0, we prove that the original Abresch--Rosenberg differential [2(H+iτ)p−(κ−4τ^2)A^2] dζ^2 is holomorphic if and only if the mean curvature H is constant; here p dζ^2 is the Hopf differential and A=⟨ξ,∂_ζ⟩. The result is local: no completeness, compactness or analyticity is assumed, and there are no exceptional surfaces. Where dH≠0, holomorphy forces the shape operator into the normal form (0&τ τ&2H) in the frame (∇ H/|∇ H|,J∇ H/|∇ H|). The round case κ=4τ^2 is settled by an elementary argument; for κ≠4τ^2 the structure equations become a rational system whose solutions are algebraic, and an embedding into a field of Puiseux series excludes every branch. In M^2(κ)×ℝ, κ≠0, where the statement fails, we show that a non-CMC surface with holomorphic original differential has nowhere vanishing dH and constant angle function, and belongs to an explicit family; in particular, closed surfaces with holomorphic original differential are CMC there. We also show that the Hopf-cylinder alternative in a theorem of Dorfmeister, Inoguchi and Kobayashi contains no non-CMC surface.
ER -