Canonical Weierstrass representations for minimal surfaces in Euclidean 4-space

TL;DR

This paper develops canonical Weierstrass representations for minimal surfaces in Euclidean 4-space, linking their geometry to holomorphic functions and establishing a correspondence with minimal surfaces in Euclidean 3-space.

Contribution

It introduces canonical parameters and explicit Weierstrass formulas for minimal surfaces in 4-space, enabling solutions to natural PDEs and geometric correspondences.

Findings

Explicit canonical Weierstrass representations derived.

Established correspondence between 4-space and 3-space minimal surfaces.

Provided applications illustrating the geometric phenomena.

Abstract

Minimal surfaces of general type in Euclidean 4-space are characterized with the conditions that the ellipse of curvature at any point is centered at this point and has two different principal axes. Any minimal surface of general type locally admits geometrically determined parameters - canonical parameters. In such parameters the Gauss curvature and the normal curvature satisfy a system of two natural partial differential equations and determine the surface up to a motion. For any minimal surface parameterized by canonical parameters we obtain Weierstrass representations - canonical Weierstrass representations. These Weierstrass formulas allow us to solve explicitly the system of natural partial differential equations and to establish geometric correspondence between minimal surfaces of general type, the solutions to the system of natural equations and pairs of holomorphic functions in the Gauss plane. On the base of these correspondences we obtain that any minimal surface of general type in Euclidean 4-space determines locally a pair of two minimal surfaces in Euclidean 3-space and vice versa. Finally some applications of this phenomenon are given.