Flows through constant mean curvature surfaces from tori to genus 2 surfaces


Lynn Heller, Sebastian Heller, Nick Schmitt

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.





Clifford torus to Lawson surface


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.





Delaunay torus to genus 2 CMC surface I






Delaunay torus to genus 2 CMC surface II