Explain what it means for an embedded surface $S$ in $\mathbf{R}^{3}$ to be minimal. What is meant by an isothermal parametrization $\phi: U \rightarrow V \subset \mathbf{R}^{3}$ of an embedded surface $V \subset \mathbf{R}^{3}$ ? Prove that if $\phi$ is isothermal then $\phi(U)$ is minimal if and only if the components of $\phi$ are harmonic functions on $U$. [You may assume the formula for the mean curvature of a parametrized embedded surface,

$$H=\frac{e G-2 f F+g E}{2\left(E G-F^{2}\right)}$$

where $E, F, G$ (respectively $e, f, g$ ) are the coefficients of the first (respectively second) fundamental forms.]

Let $S$ be an embedded connected minimal surface in $\mathbf{R}^{3}$ which is closed as a subset of $\mathbf{R}^{3}$, and let $\Pi \subset \mathbf{R}^{3}$ be a plane which is disjoint from $S$. Assuming that local isothermal parametrizations always exist, show that if the Euclidean distance between $S$ and $\Pi$ is attained at some point $P \in S$, i.e. $d(P, \Pi)=\inf _{Q \in S} d(Q, \Pi)$, then $S$ is a plane parallel to $\Pi$.