Catenoid

Example

For $u \in \R$ and $v \in \R$, let $$x(u, v) = (\cosh v \cos u, \; \cosh v \sin u, \; v).$$ Its image, the catenoid, is the surface of revolution obtained by rotating the catenary $r = \cosh z$ about the $z$-axis. It is a minimal surface.

Check. The partial derivatives are $x_u = (-\cosh v \sin u, \cosh v \cos u, 0)$ and $x_v = (\sinh v \cos u, \sinh v \sin u, 1)$. So $E = \cosh^2 v$, $F = 0$ and $G = \sinh^2 v + 1 = \cosh^2 v$, and the parametrization is conformal. For a conformal parametrization, $x_{uu} + x_{vv} = 2 E H N$, where $H$ is the mean curvature and $N$ the unit normal. Here $x_{uu} = (-\cosh v \cos u, -\cosh v \sin u, 0)$ and $x_{vv} = (\cosh v \cos u, \cosh v \sin u, 0)$, so $x_{uu} + x_{vv} = 0$ and $H = 0$.