Karcher saddle tower with $2n$ ends (Singly Periodic)

Weierstrass Data

\[ M=(\mathbb{C}\cup\{\infty\})\setminus\{z\in\mathbb{C}\;;\;z^{2n}-2\cos (n\theta)z^n+1=0\}, \] \[ g=z^{n-1},\qquad \eta = \frac{dz}{z^{2n}-2\cos (n\theta)z^n+1}, \] where $n$ is an integer greater than 1 and $\theta \in (0,\pi/(2n)]$ is a constant.

Fof each $n\ge 2$, the surface is embedded if and only if $\theta$ satisfies \[ \frac{(n-2)}{(n-1)}\le \frac{2n}{\pi}\theta\le 1. \]