A gauge theoretical generalization of Bryant's correspondence

A classical theorem in the theory of minimal surfaces establishes a correspondence between minimal surfaces in $\mathbb{R}^n$ and null holomorphic curves in $\mathbb{C}^n$. A hyperbolic version of this correspondence is due to Bryant: null holomorphic curves in ${\rm SL}(2,\mathbb{C})$ correspond to CMC-1 surfaces in the hyperbolic space $\mathbb{H}^3$. We also have a relativistic Bryant type correspondence: CMC-1 immersions in the hyperbolic space are replaced by space-like CMC-1 immersion in the de Sitter space. We prove a mutual generalisation of all these results: let $H$ be a real Lie group, $\pi:P \to M$ a principal $H$-bundle, $A$ a connection on $P$ and $\alpha\in A^1_{\rm Ad}(P,\mathfrak{h})$ a tensorial 1-form of type ${\rm Ad}$ which induces isomorphisms $A_\xi \to \mathfrak{h}$. Such a pair $(\alpha,A)$ defines an almost complex structure $J^\alpha_A$ on $P$, which is integrable if and only $(\alpha,A)$ solves a gauge-invariant first order differential system. A non-degenerate symmetric ${\rm Ad}_H$-invariant bilinear form $g$ on $\mathfrak{h}$ defines pseudo-Riemannian metrics $g^\alpha_M$, $\mathfrak{g}^\alpha_A$ on $M$, respectively $P$, and a non-degenerate bilinear form $\omega^{\alpha,g}_A:T_P\times_P T_P\to \mathbb{C}$ which is holomorphic when $J^\alpha_A$ is integrable. Assuming that this is the case, we have a Bryant type correspondence between space-like, $\omega^{\alpha,g}_A$-isotropic holomorphic immersions $Y\to P$ and space-like conformal immersions $Y\to (M,g^\alpha_M)$ whose mean curvature vector field is given by a simple explicit formula. In particular, one obtains such a correspondence for any principal bundle of the form $G\to G/H$, where $G$ is a complex Lie group, and $H$ is a real form of $G$ endowed with a non-degenerate, ${\rm Ad}_H$-invariant, symmetric bilinear form $g$ on its Lie-algebra $\mathfrak{h}$.