the surface.

#### Gauging.

Let $G : {\widetilde{\Sigma}}\longrightarrow{\Lambda_+{\mathrm{SL}}(2,{\mathbb{C}})_\rho}$ and define the new potential: $$\widehat{\xi} = \xi\cdot G := G^{-1}\xi G + G^{-1}dG.$$ The potential $\widehat{\xi}$ is a DPW potential and this operation is called “gauging”. The data $\left(\Sigma,\xi,z_0,\phi\right)$ and $\left( \Sigma,\widehat{\xi},z_0,\phi\times G(z_0) \right)$ give rise to the same immersion.

#### The monodromy problem.

Since the immersion $f$ is only defined on the universal cover ${\widetilde{\Sigma}}$, one might ask for conditions ensuring that it descends to a well-defined immersion on $\Sigma$. For any deck transformation $\tau\in{\mathrm{Deck}}\left(\widetilde{\Sigma}/\Sigma\right)$, define the monodromy of $\Phi$ with respect to $\tau$ as: $${\mathcal{M}}_\tau(\Phi)  := \Phi(\tau(z))\Phi(z)^{-1} \in {\Lambda {\mathrm{SL}}(2,{\mathbb{C}})_\rho}.$$ This map is independent of $z\in\widetilde{\Sigma}$. The standard sufficient conditions for the immersion $f$ to be well-defined on $\Sigma$ is the following set of equations, called the monodromy problem in ${\mathbb{H}}^3$: $$\label{eqMonodromyProblem}
\forall \tau\in{\mathrm{Deck}}\left(\widetilde{\Sigma}/\Sigma\right),\quad
\left\{
\begin{array}{l}
{\mathcal{M}}_\tau(\Phi)\in{\Lambda {\mathrm{SU}}(2)_\rho},\\
{\mathcal{M}}_\tau(\Phi)(e^{-q}) = \pm{\mathrm{I}}_2.
\end{array}
\right.$$ Use the point $\widetilde{z}_0$ defined in step 1 of the DPW method to identify the fundamental group $\pi_1(\Sigma,z_0)$ with ${\mathrm{Deck}}(\widetilde\Sigma / \Sigma)$. Let $\left\{ \gamma_i \right\}_{i\in I}$ be a set of generators of $\pi_1\left(\Sigma,z_0\right)$. Then the problem is equivalent to $$\label{eqMonodromyProblemLoop}
\forall i\in I, \quad 
\left\{
\begin{array}{l}
{\mathcal{M}}_{\gamma_i}(\Phi)\in{\Lambda {\mathrm{SU}}(2)_\rho},\\
{\mathcal{M}}_{\gamma_i}(\Phi)(e^{-q}) = \pm{\mathrm{I}}_2.
\end{array}
\right.$$

#### Example: 