Surfaces of constant mean curvature in the Heisenberg group (Q2880505)

The author studies the surfaces of constant mean curvature in the three-dimensional Heisenberg group Nil. The main result reads that every surface of constant mean curvature \(H\) in the neighborhood of a non-umbilical point corresponds to a solution \(v=\rho+i\varphi\) of the following system of two equations NEWLINE\[NEWLINE v_{z\bar{z}}+2\sinh 2v =0, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \frac{\varphi_x^2}{(\cosh\rho)^2}+ \frac{\varphi_y^2}{(\sinh\rho)^2}= 8\biggl(\cos 2\rho -\text{Re\,}\frac{2H+i}{2H-i}\biggr). NEWLINE\]NEWLINE The study is based on the Weierstrass representation of surfaces in the three-dimensional Lie groups proposed in [\textit{I. A. Tajmanov}, Tr. Mat. Inst. Steklova 244, 249--280 (2004; Zbl 1091.53041)].