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CMC Surfaces with Branch Points

Branch points (where the derivative has rank 0) are the only type of singularity for CMC surfaces in Euclidean 3-space. The first image is the simplest type, with DPW potential off-diag(z,1)λ1dz. The other singularity shown is a non-generic singularity of a spacelike CMC surface in Minkowki 3-space, produced by the solution of the singular Björling problem with Björling data s(x)=x, t(x)=x and θ(x)=0.01. It is like a branched swallowtail.