If $U$ denotes a domain in $\mathbb{R}^{2}$, what is meant by saying that a smooth map $\phi: U \rightarrow \mathbb{R}^{3}$ is an immersion? Define what it means for such an immersion to be isothermal. Explain what it means to say that an immersed surface is minimal.

Let $\phi(u, v)=(x(u, v), y(u, v), z(u, v))$ be an isothermal immersion. Show that it is minimal if and only if $x, y, z$ are harmonic functions of $u, v$. [You may use the formula for the mean curvature given in terms of the first and second fundamental forms, namely $\left.H=(e G-2 f F+g E) /\left(2\left\{E G-F^{2}\right\}\right) .\right]$

Produce an example of an immersed minimal surface which is not an open subset of a catenoid, helicoid, or a plane. Prove that your example does give an immersed minimal surface in $\mathbb{R}^{3}$.