(i) Define what is meant by an isothermal parametrization. Let $\phi: U \rightarrow \mathbb{R}^{3}$ be an isothermal parametrization. Prove that

$$\phi_{u u}+\phi_{v v}=2 \lambda^{2} \mathbf{H}$$

where $\mathbf{H}$ is the mean curvature vector and $\lambda^{2}=\left\langle\phi_{u}, \phi_{u}\right\rangle$.

Define what it means for $\phi$ to be minimal, and deduce that $\phi$ is minimal if and only if $\Delta \phi=0$.

[You may assume that the mean curvature $H$ can be written as

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

(ii) Write $\phi(u, v)=(x(u, v), y(u, v), z(u, v))$. Consider the complex valued functions

$$\varphi_{1}=x_{u}-i x_{v}, \quad \varphi_{2}=y_{u}-i y_{v}, \quad \varphi_{3}=z_{u}-i z_{v}$$

Show that $\phi$ is isothermal if and only if $\varphi_{1}^{2}+\varphi_{2}^{2}+\varphi_{3}^{2} \equiv 0$.

Suppose now that $\phi$ is isothermal. Prove that $\phi$ is minimal if and only if $\varphi_{1}, \varphi_{2}$ and $\varphi_{3}$ are holomorphic functions.

(iii) Consider the immersion $\phi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ given by

$$\phi(u, v)=\left(u-u^{3} / 3+u v^{2},-v+v^{3} / 3-u^{2} v, u^{2}-v^{2}\right)$$

Find $\varphi_{1}, \varphi_{2}$ and $\varphi_{3}$. Show that $\phi$ is an isothermal parametrization of a minimal surface.