Spacelike surfaces with harmonic inverse mean curvature (Q2712595)

Following papers of A.Bobenko and his collaborators on surfaces with harmonic inverse mean curvature in 3-dimensional Euclidean space the authors consider a natural generalization of this problem, namely spacelike surfaces with harmonic inverse mean curvature (SHIMC) in 3-dimensional Lorentzian space. The paper is self-contained and written in a very clear, well-ordered way. It starts with a detailed introduction to relevant topics of semi-Riemannian (or pseudo-Riemannian) geometry: split-quaternion formalism and spacelike surfaces in Lorentzian 3-spaces of constant curvature (i.e., in the Minkowski 3-space, de Sitter 3-space and anti de Sitter 3-space). NEWLINENEWLINENEWLINEThe Lax pair is postulated in the standard way (in full analogy to the constant mean curvature surfaces case but with the spectral parameter depending on the coordinates). The compatibility conditions imply that the inverse of the mean curvature is harmonic. The corresponding immersion in the Minkowski space is obtained using the Sym formula. The formulas for immersions in de Sitter and anti de Sitter spaces are given as well. Special attention is given to surfaces of rotation. NEWLINENEWLINENEWLINEThen the authors present detailed results on Bonnet surfaces (which, by definition, admit deformations preserving principal curvatures). The Bonnet surfaces are dual to (isothermic) SHIMC surfaces. The ordinary differential equation (generalized Hazzidakis equation) describing Bonnet surfaces is discussed. As a rule, the obtained results turn out to be rather similar to those known earlier in the Euclidean case.