Willmore Surfaces of Revolution

Willmore Surfaces of Revolution.

Surfaces of revolution in ${\mathbb S}^3$ are those invariant under the action of a 1-parameter subgroup that fixes a geodesic. Under stereographic projection from an appropriate point they project to surfaces of revolution in ${\mathbb R}^3$, as in the first three images shown here. The surfaces here are all stereographic projections of Willmore surfaces of revolution in the 3-sphere, and therefore they are also Willmore surfaces in ${\mathbb R}^3$. The first image is (an inversion of) a catenoid. The others are conformally congruent to minimal surfaces in ${\mathbb H}^3$. Read about this work...