Below are depicted three flows through constant mean curvature (CMC) surfaces. Each flow starts from a CMC torus, which is “opened up” at four fixed point of an order-2 symmetry. During the flow, the angle at these points decreases until it is 2/3 of a circle. Filling in the missing 1/3 inserts a handle, completing a genus 2 surface. Continuing the flow generates CMC surfaces of arbitrarily high genus.
The first sequence (Clifford to Lawson) is a flow through minimal surfaces; the remaining two are flows through CMC surfaces. These last two differ in the choice of order-2 symmetry of the initial Delaunay torus.
Each sequence shows four snapshots of the flow, at which the continuous genus parameter g takes values g=1, 4/3, 5/3, 2. Two views of the same surface are shown at each snapshot. The last two rows in each sequence show the same final genus 2 surface before and after the missing piece is filled in.
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Flow through ξg,1 from the Clifford torus ξ1,1 to Lawson’s surface ξ2,1. The column on the far right shows the Plateau solution for a piece of the surface.
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