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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)λ−1dz.
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.