Möbius-type minimal surfaces family in R4${\mathbb{R}}^{4}$ via the generalized Weierstrass–Enneper representation

Abstract In this paper, we study the Möbius-type minimal surfaces family obtained via the generalized Weierstrass–Enneper representation of minimal surfaces in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:msup> </m:math> ${\mathbb{R}}^{4}$ . We establish the holomorphic nullity condition, prove conformality and minimality, and construct explicit parametrizations for specific choices of holomorphic data. We analyze two-dimensional orthogonal projections of these surfaces and compute an orthonormal basis of two normal vectors using the Gram-Schmidt process. We also compare this representation with its <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> </m:math> ${\mathbb{R}}^{3}$ analogue and provide geometric insights into the induced metric, Gauss map, curvature, and the structure of holomorphic null curves in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">C</m:mi> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:msup> </m:math> ${\mathbb{C}}^{4}$ . In particular, we highlight the role of the Grassmannian G (2, 4) in describing the Gauss map and discuss how the rank-2 normal bundle influences curvature computations. Explicit examples derived from polynomial data illustrate the geometric richness that arises in codimension two. These developments collectively provide a coherent analytic and geometric framework for understanding minimal surfaces in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:msup> </m:math> ${\mathbb{R}}^{4}$ and for extending classical Weierstrass theory into higher dimensions.