The gallery above shows the evolution of a doubled helicoid, thrice punctured, during regeneration. These surfaces are invariant under a screw motion, have genus 2 in the quotient, and four ends. The screw motion angles in the frames are 90º, 144º, 162º, 178º, respectively.
We have a local existence and embeddedness theorem for the 2-helicoid case, in which the regenerated surfaces are assumed to be invariant under a hyperelliptic involution that exchanges the two helicoids. The proofs are AI-assisted and will be made available shortly.
The non-hyperelliptic case is open. Some non-degenerate nodal configurations do not regenerate to embedded surfaces.
Also open is the question of whether one can regenerate from more than two helicoids.
Above are two regenerated surfaces originating from two different 4-node configurations.
Not much is known about the nature of the solutions of these balance equations for the nodes. Configurations with any number of collinear nodes are known to exist; below is such an example with six nodes:
It is generally unknown where the long-term evolution of these surfaces leads. The 3-noded case on top appears to converge to two separating singly periodic genus one helicoids. This is a conjecture. The 2-noded case converges provably to a noded configuration of planes involving one catenoid and two helicoids with equal spin. The more complicated cases are completely open.
The research notes will follow soon.
