Three planar ends between two catenoid ends (finite Riemann 3)

Weierstrass Data

\[ M=\mathbb{C}\setminus\{\pm 1, \pm a\}, \] \[ g=\frac{z^4-2bz^2+c}{\rho (z^2-a^2)(z^2+3)},\qquad \eta = \left(\frac{z^2+3}{z^2-1}\right)^2dz, \] where $\rho\in (0,\sqrt{3})$ is a constant, and $a$, $b$, $c$ are positive constants depend on $\rho$: \begin{align*} a&=\sqrt{\frac{7\rho^2+1+2\sqrt{12\rho^4+4\rho^2+1}}{3-\rho^2}}, \\ b&=\frac{-6\rho^4+3+\rho^2(6-\sqrt{12\rho^4+4\rho^2+1})}{3-\rho^2}, \\ c&=\frac{20\rho^4+3+\rho^2(5+6\sqrt{12\rho^4+4\rho^2+1})}{3-\rho^2}. \end{align*}