Minimal surfaces in the soliton surface approach

TL;DR

This paper derives the classical Enneper-Weierstrass representation of minimal surfaces using the soliton surface approach, connecting hyperbolic space representations, linear problems, and differential equations.

Contribution

It introduces a novel limiting procedure to recover the Enneper-Weierstrass formula from Bryant-type representations in hyperbolic space.

Findings

Derived Enneper-Weierstrass representation via soliton surface approach.

Established connection between Bryant representation and second-order linear ODEs.

Applied the method to the error function equation as an example.

Abstract

The main objective of this paper is to derive the Enneper-Weierstrass representation of minimal surfaces in $\mathbb{E}^3$ using the soliton surface approach. We exploit the Bryant-type representation of conformally parametrized surfaces in the hyperbolic space $H^3(\lambda)$ of curvature $-\lambda^2$, which can be interpreted as a 2 by 2 linear problem involving the spectral parameter $\lambda$. In the particular case of constant mean curvature-$\lambda$ surfaces a special limiting procedure $(\lambda\rightarrow 0)$, different from that of Umehara and Yamada [33], allows us to recover the Enneper-Weierstrass representation. Applying such a limiting procedure to the previously known cases, we obtain Sym-type formulas. Finally we exploit the relation between the Bryant representation of constant mean curvature-$\lambda$ surfaces and second-order linear ordinary differential equations. We illustrate this approach by the example of the error function equation.