Weierstrass-type constructions, variational analysis and integral-free minimal immersions in $\mathbb{R}^{n}$

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Abstract

We study conformal minimal immersions into Rⁿ via the classical correspondence with holomorphic null curves in Cⁿ. After recalling a convenient Weierstrass-type parametrization of null data on simply connected domains, we present an explicit integral-free construction that produces minimal immersions directly from a single holomorphic seed function, meaning that the coordinate functions are obtained through closed-form algebraic expressions involving derivatives of the seed rather than through path integration of null data. This viewpoint leads to a reconstruction identity for the seed and gives concrete formulas for the induced metric and the associated Gauss map. For polynomial seeds, we obtain explicit families in arbitrary codimension with closed-form conformal factors. As an analytic application, we derive a second-variation formula for the area under holomorphic perturbations of the seed, expressed in terms of the third derivative of the perturbation. The discussion is local and formula-driven: we do not consider global period problems, completeness, embeddedness, or topological classification, as our goal is to develop explicit analytic constructions rather than a global classification theory.

Keywords

minimal immersions / holomorphic null curves / Weierstrass-type formulas / conformal parametrizations / integral-free construction / second variation of area

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