Minimal surfaces associated with orthogonal polynomials

Abstract

This paper is devoted to a study of the connection between the immersion functions of two-dimensional surfaces in Euclidean or hyperbolic spaces and classical orthogonal polynomials. After a brief description of the soliton surfaces approach defined by the Enneper-Weierstrass formula for immersion and the solutions of the Gauss-Weingarten equations for moving frames, we derive the three-dimensional numerical representation for these polynomials. We illustrate the theoretical results for several examples, including the Bessel, Legendre, Laguerre, Chebyshev and Jacobi functions. In each case, we generate a numerical representation of the surface using the Mathematica symbolic software.

Keywords:

• Integrable system
• soliton surface
• minimal surface
• orthogonal polynomial
• Weierstrass immersion formula
• Sym-Tafel immersion formula
• CMC surface

2000 Mathematics Subject Classification:

• 53A07
• 53B50
• 53C43
• 81T45 PACS numbers: 02.40.-k
• 02.40.Hw
• 02.40.Ma