Wei's surface (doubly periodic)

Weierstrass Data

\[ \overline{M}=\left\{(z,w)\in(\mathbb{C}\cup\{\infty\})^2\;;\;w^2=\frac{(z-\lambda)(z+1)(z-\lambda_1)}{(z+\lambda)(z-1)(z+\lambda_1)}\right\}, \] where $\lambda$ and $\lambda_1$ are real constants so that $0\lt\lambda\lt 1\lt\lambda_1$. \[ M=\overline{M}\setminus\{(a,1),\,(a,-1),\,(-a,1),\,(-a,-1)\}, \] where $a=\sqrt{\lambda\lambda_1/(\lambda +\lambda_1 -1)}$. \[ g=w,\qquad \eta = \frac{dz}{(z^2-a^2)w}. \] For given $\lambda\in (0,1)$, we can choose $\lambda_1\gt 1$ so that $f$ is single valued on $M$ as a map into $\mathbb{R}^3/\Lambda$, where $\Lambda=\{n_1(v_1,0,0)+n_2(0,v_2,0)\;;\;n_1,n_2\in\mathbb{Z}\}$ for some non-zero real constants $v_1$, $v_2$.