Complex Quaternions and Superminimal Surfaces in Four-Space

Abstract

We develop a quaternionic approach to conformal superminimal surfaces in Euclidean four-space. The starting point is the classical Weierstrass representation: every conformal minimal immersion $X\colon M\to\mathbb{R}^4$ is recovered as $X = c + Re\int\Phi dz$, where $\Phi$ is a holomorphic null curve in $\mathbb{C}^4$, identified with the algebra of complex quaternions $\mathbb{H}\otimes\mathbb{C}$. The multiplicativity of the quaternionic symmetrized norm makes it natural to factor null curves as $\Phi=ALB$, where $A$ and $B$ are holomorphic maps of unit symplectic norm and $L$ is a holomorphic null element. We show that on simply connected domains the null factor can always be taken constant. The main result is an explicit quaternionic reformulation of the superminimality condition -- the requirement that the curvature ellipse be a circle at every point. In the fixed-null gauge $L=1+\sqrt{-1} e_1$, superminimality is equivalent to the vanishing of a product of two holomorphic functions built from the left and right Maurer--Cartan forms of $A$ and $B$. On a connected domain, this forces one of the two components of the generalized Gauss map $[\Phi]\colon M\to Q^2\simeq\mathbb{CP}^1\times\mathbb{CP}^1$ to be constant, recovering in spinorial terms the classical ruling condition on the projective null quadric. We further provide a first-order ODE parametrization of the superminimal $ALB$-data, analyse the residual gauge freedom, prove a fixed-gauge rigidity statement for polynomial spinorial factors, and illustrate the theory with explicit examples.