We say that a parametrization $\phi: U \rightarrow S \subset \mathbf{R}^{3}$ of a smooth surface $S$ is isothermal if the coefficients of the first fundamental form satisfy $F=0$ and $E=G=\lambda(u, v)^{2}$, for some smooth non-vanishing function $\lambda$ on $U$. For an isothermal parametrization, prove that

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

where $\mathbf{N}$ denotes the unit normal vector and $H$ the mean curvature, which you may assume is given by the formula

$$H=\frac{g+e}{2 \lambda^{2}}$$

where $g=-\left\langle\mathbf{N}_{u}, \phi_{u}\right\rangle$ and $e=-\left\langle\mathbf{N}_{v}, \phi_{v}\right\rangle$ are coefficients in the second fundamental form.

Given a parametrization $\phi(u, v)=(x(u, v), y(u, v), z(u, v))$ of a surface $S \subset \mathbf{R}^{3}$, we consider the complex valued functions on $U$ :

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

Show that $\phi$ is isothermal if and only if $\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}=0$. If $\phi$ is isothermal, show that $S$ is a minimal surface if and only if $\theta_{1}, \theta_{2}, \theta_{3}$ are holomorphic functions of the complex variable $\zeta=u+i v$

Consider the holomorphic functions on $D:=\mathbf{C} \backslash \mathbf{R}_{\geqslant 0}$ (with complex coordinate $\zeta=u+i v$ on $\mathbf{C})$ given by

$$\theta_{1}:=\frac{1}{2}\left(1-\zeta^{-2}\right), \quad \theta_{2}:=-\frac{i}{2}\left(1+\zeta^{-2}\right), \quad \theta_{3}:=-\zeta^{-1}$$

Find a smooth map $\phi(u, v)=(x(u, v), y(u, v), z(u, v)): D \rightarrow \mathbf{R}^{3}$ for which $\phi(-1,0)=\mathbf{0}$ and the $\theta_{i}$ defined by (2) satisfy the equations (1). Show furthermore that $\phi$ extends to a smooth map $\tilde{\phi}: \mathbf{C}^{*} \rightarrow \mathbf{R}^{3}$. If $w=x+i y$ is the complex coordinate on $\mathbf{C}$, show that

$$\widetilde{\phi}(\exp (i w))=(\cosh y \cos x+1, \cosh y \sin x, y)$$