Surfaces of revolution with periodic mean curvature (Q1411276)

This report studies surfaces of revolution with periodic mean curvature as an extension of the CMC (constant mean curvature) surface theory. It contains two main results. The first one represents a criterion for a periodic function to be the mean curvature of a periodic surface of revolution. The second one gives a construction method for those periodic surfaces of revolution whose mean curvatures are periodic functions that satisfy the criterion. The reviewer finds the result significant and of interest to anybody who studies CMC surfaces and related areas. An interesting remark is that the surface associated with a plane closed curve has a 1-parameter family of such periodic surfaces with the same mean curvature function, changing the starting point, and this indeed is similar to the deformation from unduloid to nodoid from CMC surfaces. The reviewer would be interested in knowing the description of the representations that surfaces of revolution with periodic mean curvature admit. Also, if there are explicit analogouses of the normalized (holomorphic) potentials that recently were found for CMC surfaces, for the case of surfaces of revolution with periodic mean curvature.