Elementary proof of a Lemma for parallel mean curvature surfaces in complex space forms
Katsuei Kenmotsu
TL;DR
This work delivers an elementary proof of Kenmotsu's Lemma for parallel mean curvature surfaces in complex space forms, showing that when the ambient curvature parameter $\rho$ is nonzero and the Kaehler angle $\alpha$ is not constant, a portion of the complexified second fundamental tensor is determined by $\alpha$. The authors replace exterior calculus with a coordinate-based approach, using $\alpha$ as a local coordinate (with a companion coordinate $\beta$) and partial differentiation to derive structure equations for the second fundamental form components $a$ and $c$. They prove that for general-type immersions, $a$ is a function of $\alpha$, aided by a nontrivial polynomial relation for $a_\alpha$ and a Mathematica-verified obstruction $G(\pi/4,0,0)\neq 0$, thus supporting a local classification program. The work provides a computational framework via explicit $p_i(\alpha,a,\bar a)$ in the Appendix to validate the key nontriviality condition, strengthening the understanding of the local geometry of such surfaces.
Abstract
We provide an elementary proof of a lemma that plays an important role in the classification of parallel mean curvature surfaces in two-dimensional complex space forms.