Enneper surface

Example

For $(u, v) \in \R^2$, let $$x(u, v) = \left(u - \frac{u^3}{3} + u v^2, \; -v + \frac{v^3}{3} - u^2 v, \; u^2 - v^2\right).$$ Its image is the Enneper surface. It is a minimal surface.

Check. The partial derivatives are $x_u = (1 - u^2 + v^2, \, -2uv, \, 2u)$ and $x_v = (2uv, \, -1 + v^2 - u^2, \, -2v)$. A direct computation gives $E = G = (1 + u^2 + v^2)^2$ and $F = 0$, so $x$ is a conformal immersion. The second derivatives are $x_{uu} = (-2u, -2v, 2)$ and $x_{vv} = (2u, 2v, -2)$, so $x_{uu} + x_{vv} = 0$. For a conformal parametrization this means that the mean curvature vanishes.