Henneberg-Type Minimal Immersions in ${\mathbb{R}}^4$

Abstract

We construct a family of higher-order Henneberg-type minimal surfaces in ${\mathbb{R}}^4$ from explicit holomorphic data in a generalized Weierstrass--Enneper representation. The construction retains the polynomial reciprocal structure of the classical Henneberg data and yields explicit holomorphic primitives and real parametrizations for the family ${\mathcal{H}}_{(m,n)}$. We distinguish regular points from branch points, analyze the role of the complex parameter $\lambda$, and show that the real-$\lambda$ subfamily ${\mathcal{H}}_{(1,1)}$ lies in a three-dimensional linear subspace. For ${\mathcal{H}}_{(1,1)}$, we compute the induced conformal metric and Gaussian curvature, proving that the curvature is negative at every regular point. An explicit parametrization is given for ${\mathcal{H}}_{(1,1)}$, together with three-dimensional coordinate projections used solely for visualization. We also derive an integral-free formulation in terms of a single holomorphic seed function and identify the exceptional values $\lambda =\pm i$ for the inverse reconstruction formula.

arXiv comment14 pages, 2 figures