On the periodicity of planes with parallel mean curvature vector in \(\mathbb{C} H^2\)
The paper studies isometric immersions of the Euclidean plane in the complex hyperbolic plane \(\mathbb C H^2\) with non-zero parallel mean curvature vector, and gives explicit conditions under which they are invariant under an action of \(\mathbb Z\times \mathbb Z\) or of \(\mathbb Z\). The proof (and actually the statement of the main result) uses a description of such parallel mean curvature isometric immersions which was given by \textit{K. Kenmotsu} and \textit{D. Zhou} [Am. J. Math. 122, 295--317 (2000; Zbl 0997.53006)]. They showed that such immersions are characterized by two numbers, one of which is the length of the mean curvature vector and the other one is defined in a less simple way in terms of two quadratic holomorphic differentials associated to the immersion. The paper under review shows that some inequalities on those two numbers determine whether the immersion is periodic.
scientific article; zbMATH DE number 2145307