The prevalence of tori amongst constant mean curvature planes in R3R3

Constant mean curvature (CMC) tori in Euclidean 3-space are described by an algebraic curve, called the spectral curve, together with a line bundle on this curve and a point on S1S1, called the Sym point. For a given spectral curve the possible choices of line bundle and Sym point are easily described. The space of spectral curves of tori is totally disconnected. Hence to characterise the “moduli space” of CMC tori one should, for each genus gg, determine the closure Pg¯ of spectral curves of CMC tori within the spectral curves of CMC planes having spectral genus gg. We identify a real subvariety RgRg and a subset Sg⊆RgSg⊆Rg such that Rmax  g⊆Pg¯⊆Sg, where Rmax  g denotes the points of RgRg having maximal dimension. The lowest spectral genus for which tori exist is g=2g=2 and in this case R2=Rmax2=P2¯=S2. For g>2g>2, we conjecture that Rg⊋Rmaxg=Sg. We give a number of alternative characterisations of Rmaxg and in particular introduce a new integer invariant of a CMC plane of finite type, called its winding number.