Explicit Minimal Surface Models in $\mathbb{R}^5$ via Holomorphic Null Curves

TL;DR

This paper develops explicit formulas and constructions for conformal minimal immersions into five-dimensional Euclidean space using holomorphic null curves, emphasizing computational and visualization aspects.

Contribution

It isolates the five-dimensional case, providing explicit, integral-free formulas, polynomial seed analysis, and connections to moving frames and integrable systems.

Findings

Derived a family of minimal immersions depending on a seed function and parameters

Explicit formulas for immersion and metric in closed form

Analyzed polynomial seeds and their geometric properties

Abstract

We study explicit conformal minimal immersions into $\mathbb{R}^5$ obtained from holomorphic null curves in $\mathbb{C}^5$. Although the general correspondence between conformal minimal immersions in $\mathbb{R}^n$ and holomorphic null data in $\mathbb{C}^n$ is classical, our aim here is different. We isolate the five-dimensional case and develop a concrete, self-contained account that emphasizes explicit formulas, integral-free constructions, and coordinate expressions suitable for computation and visualization. Starting from a Weierstrass-type representation in $\mathbb{R}^5$, we derive a family of conformal minimal immersions depending on a single holomorphic seed function and two real parameters. The resulting formulas allow the immersion and the induced metric to be written in closed form. We then examine polynomial seeds in detail, derive their polar and Cartesian expansions, and discuss the geometric information carried by natural coordinate projections. We reinterpret the construction in the language of moving frames, the generalized Gauss map, and a local DPW-type scheme. This provides a conceptual bridge between explicit holomorphic formulas and the Cartan-integrable-systems viewpoint. The discussion is local and formula-driven; global questions such as periods, completeness, and embeddedness lie beyond the present scope. We also briefly clarify why the complex-analytic structure underlying the representation is essential, and why it cannot be replaced by a naive quaternionic formalism, due to the loss of commutativity, holomorphic structure, and compatibility with the null-curve framework.