Two-ended surfaces with least total absolute curvature by Fujimori and Shoda

Weierstrass Data

\[ \overline{M}=\left\{(z,w)\in(\mathbb{C}\cup\{\infty\})^2\;;\;w^{k+1}=z^2\left(\frac{z-1}{z-a}\right)^k\right\}, \] where $k$ is positive even integer and $a$ is a constant so that \( a > 1 \). \[ M=\overline{M}\setminus\{(0,0),\,(\infty ,\infty)\}, \] \[ g=a^{(k-2)/(2k+2)}w,\qquad \eta = \frac{dz}{zw}. \] For given $k$, we can choose $a$ so that $f$ is single valued on $M$.

The surface is of genus $k$ with 2 ends. Each end is non-embedded and asymptotic to the double cover of the catenoid. The intersection of the surface and $xy$-plane looks like an epitrochoid, but in fact it is not.