New surfaces of constant mean curvature (Q1323426)

Examples of surfaces in \(R^ 3\) with constant mean curvature \(H = 1\) have been constructed by many authors using different approaches. In the present work the author extends a conjugate surface construction of Karcher which is based on ideas of Lawson to determine a wealth of highly symmetric families of \(H\)-surfaces with ends. In the conjugate surface construction a fundamental piece for the symmetry group of an \(H\)-surface in \(R^ 3\) is constructed by solving a Plateau problem in \(S^ 3\) for an appropriate polygonal contour. The paper explains the construction in detail and uses it to prove existence of various examples having compact or infinitely long contours. Existence of infinitely long fundamental pieces is carefully studied.