Costa-Hoffman-Meeks surface of genus $k$

Weierstrass Data

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