Only constant mean curvature surfaces in $E(κ,τ)$, $τ\ne0$, have a holomorphic original Abresch-Rosenberg differential

Abstract

For immersed surfaces in the homogeneous spaces $E(κ,τ)$ with $τ\ne0$, 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 $A=\langleξ,\partial_ζ\rangle$. The result is local: no completeness, compactness or analyticity is assumed, and there are no exceptional surfaces. Where $dH\ne0$, holomorphy forces the shape operator into the normal form $\bigl(\begin{smallmatrix}0&τ\\τ&2H\end{smallmatrix}\bigr)$ in the frame $(\nabla H/|\nabla H|,J\nabla H/|\nabla H|)$. The round case $κ=4τ^2$ is settled by an elementary argument; for $κ\ne4τ^2$ the structure equations become a rational system whose solutions are algebraic, and an embedding into a field of Puiseux series excludes every branch. In $M^2(κ)\times\mathbb{R}$, $κ\ne0$, 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.