Cylindrical surfaces of Delaunay (Q2368613)

Classical Delaunay surfaces of revolution of constant mean curvature 1 are of the four types: nodoid, spheres, unduloid, and cylinder. In this paper, instead of the usual area, the author considers the cylindrical norm \(\Phi\) on \(\mathbb R^ 3\) which is the sum of areas of horizontal and vertical projections, for which the isoperimetric shape is a cylinder rather than a round ball. For this norm, the author provides analogous cylindrical surfaces of Delaunay. It is shown that for the cylindrical shape there are precisely two one-parameter families of Delaunay surfaces of revolution of generalized constant mean curvature 1. The unduloids are embedded chains of similar cylinders of radius alternating between \(R\) and \(r < R\) and height alternating between \(H\) and \(h < H\), with \(R + r = 1\) and \(\frac{H}{R}=\frac{h}{r}=\frac2{R-r}\). The nodoids are immersed chains of similar cylinders of alternating orientation, radius \(R\) or \(r < R\), height \(H\) or \(h < H\), with \(R - r = 1\) and \(\frac{H}{R}=\frac{h}{r}=\frac2{R+r}\). There are no other connected complete piecewise smooth surfaces of revolution of generalized mean curvature 1.