@article{Güler2026, 
author = {Erhan Güler and Magdalena Toda},
title = {Weierstrass-type constructions, variational analysis and integral-free minimal immersions in              R        n},
year = {2026},
journal = {Electronic Research Archive},
volume = {34},
number = {3},
pages = {1885-1899},
keywords = {minimal immersions, holomorphic null curves, Weierstrass-type formulas, conformal parametrizations, integral-free construction, second variation of area},
url = {https://www.sciopen.com/article/10.3934/era.2026084},
doi = {10.3934/era.2026084},
abstract = {We study conformal minimal immersions into              R        n   via the classical correspondence with holomorphic null curves in              C        n  . 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.}
}