In this study, we present a generalized Weierstrass-Enneper representation for minimal surfaces in four-dimensional Euclidean space. We derive both parametric (explicit) and algebraic (implicit) representations of several example minimal surfaces, examine their differential-geometric properties, and visualize them through orthogonal projections from ${\mathbb{R}}^4$ into ${\mathbb{R}}^3$.

We study conformal minimal immersions into ${\mathbb{R}}^n$ via the classical correspondence with holomorphic null curves in ${\mathbb{C}}^n$. 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.

Red blood cells (RBCs) are classically modeled as thin elastic surfaces governed by the Helfrich–Canham energy. Within the widely used axisymmetric framework, we provide what appears to be the first complete mathematical proof that Cassini ovals, except for the limiting round sphere, do not satisfy the Helfrich shape equation, even though they can approximate biconcave profiles over experimentally relevant parameter ranges. The argument proceeds by direct substitution of the Cassini meridian into the third-order reduced shape equation and yields an over-determined algebraic system with no consistent solution for any nonzero Cassini eccentricity.

Beyond this structural result, we place the analysis in a broader biophysical and geometric context. We review typical bending and shear moduli, reduced-volume constraints, and the role of area and volume conservation; we explain the geometric meaning of the Helfrich parameters and their relation to the Willmore functional; we connect our formulation to bilayer-couple and area-difference-elasticity (ADE) models; and we outline practical routes to parameter inference from micropipette aspiration, flicker spectroscopy, and optical tweezers. We also discuss why the specific curvature structure of Cassini ovals is conceptually incompatible with Helfrich equilibria, indicate regimes in which Cassini profiles remain useful surrogates for geometric descriptors and for initializing PDE-based solvers, and summarize recent models based on constant bending-energy density. Finally, we identify extensions with spatially varying spontaneous curvature that may accommodate membrane heterogeneity and more complex RBC morphologies.