(i) What is a minimal surface? Explain why minimal surfaces always have non-positive Gaussian curvature.

(ii) A smooth map $f: S_{1} \rightarrow S_{2}$ between two surfaces in 3-space is said to be conformal if

$$\left\langle d f_{p}\left(v_{1}\right), d f_{p}\left(v_{2}\right)\right\rangle=\lambda(p)\left\langle v_{1}, v_{2}\right\rangle$$

for all $p \in S_{1}$ and all $v_{1}, v_{2} \in T_{p} S_{1}$, where $\lambda(p) \neq 0$ is a number which depends only on $p$.

Let $S$ be a surface without umbilical points. Prove that $S$ is a minimal surface if and only if the Gauss map $N: S \rightarrow S^{2}$ is conformal.

(iii) Show that isothermal coordinates exist around a non-planar point in a minimal surface.