We introduce and study a class of Scherk-type minimal surfaces immersed in the four-dimensional Euclidean space $ \mathbb{R}^4 $. Motivated by the classical Scherk minimal surface in $ \mathbb{R}^3 $, we construct higher-codimensional analogues using the generalized Weierstrass–Enneper representation for minimal surfaces in $ \mathbb{R}^4 $. Explicit parametrizations are derived from holomorphic null curves in $ \mathbb{C}^4 $, ensuring conformality and vanishing mean curvature. The geometric properties of the resulting surface are examined through real parametrizations, orthogonal projections, and the explicit construction of an orthonormal frame for the normal bundle. Representative special cases are presented to illustrate how the geometric structure characteristic of a Scherk surface extends naturally to higher codimension.
Citation: Magdalena Toda, Erhan Güler. On Scherk-type minimal immersion in $ \mathbb{R}^4 $ constructed by the generalized Weierstrass–Enneper representation[J]. AIMS Mathematics, 2026, 11(3): 5456-5475. doi: 10.3934/math.2026225
We introduce and study a class of Scherk-type minimal surfaces immersed in the four-dimensional Euclidean space $ \mathbb{R}^4 $. Motivated by the classical Scherk minimal surface in $ \mathbb{R}^3 $, we construct higher-codimensional analogues using the generalized Weierstrass–Enneper representation for minimal surfaces in $ \mathbb{R}^4 $. Explicit parametrizations are derived from holomorphic null curves in $ \mathbb{C}^4 $, ensuring conformality and vanishing mean curvature. The geometric properties of the resulting surface are examined through real parametrizations, orthogonal projections, and the explicit construction of an orthonormal frame for the normal bundle. Representative special cases are presented to illustrate how the geometric structure characteristic of a Scherk surface extends naturally to higher codimension.
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Magdalena Toda, Erhan Güler. On Scherk-type minimal immersion in $ \mathbb{R}^4 $ constructed by the generalized Weierstrass–Enneper representation[J]. AIMS Mathematics, 2026, 11(3): 5456-5475. doi: 10.3934/math.2026225
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