Willmore Surfaces of Revolution: Profile Curves.
Rotational Willmore surfaces in ${\mathbb S}^3$ can be described as a 2-parameter family, with real parameters $(m,\beta)$.
Representative samples are obtained by taking $m=0$ and varying $\beta$. Then the surface is conformally congruent to
a minimal surface in ${\mathbb S}^3$ if and only if $|\beta|<1$, to a minimal surface in ${\mathbb R}^3$ if and only if
$|\beta|=1$, and otherwise to a minimal surface in ${\mathbb H}^3$.
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