Let $U$ be a domain in $\mathbb{R}^{2}$, and let $\phi: U \rightarrow \mathbb{R}^{3}$ be a smooth map. Define what it means for $\phi$ to be an immersion. What does it mean for an immersion to be isothermal?

Write down a formula for the mean curvature of an immersion in terms of the first and second fundamental forms. What does it mean for an immersed surface to be minimal? Assume that $\phi(u, v)=(x(u, v), y(u, v), z(u, v))$ is an isothermal immersion. Prove that it is minimal if and only if $x, y, z$ are harmonic functions of $u, v$.

For $u \in \mathbb{R}, v \in[0,2 \pi]$, and smooth functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$, assume that

$$\phi(u, v)=(f(u) \cos v, f(u) \sin v, g(u))$$

is an isothermal immersion. Find all possible pairs $(f, g)$ such that this immersion is minimal.