SO(1,3)-equivariant Willmore surfaces in hyperbolic 3-space

SO(1,3)-equivariant Willmore surfaces in hyperbolic 3-space.

The first three surfaces are Moebius equivalent, as are the last two. Each of these images can be interpreted either as the Poincare ball image of a Willmore surface in ${\mathbb H}^3$, or as a stereographic projection to ${\mathbb R}^3$ of a Moebius equivalent surface in ${\mathbb S}^3$, or simply as a Willmore surface in ${\mathbb R}^3$. The last surface (Image 6) is in fact a minimal surface in ${\mathbb R}^3$ that contains the logarithmic spiral $e^t(\cos t, \sin t)$. Read about this work...