Special Functions on the Sphere with Applications to Minimal Surfaces

✍ Scribed by Frank Baginski

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 438 KB
• Volume: 28
• Category: Article
• ISSN: 0196-8858
• DOI: 10.1006/aama.2001.0788

✦ Synopsis

A function which is homogeneous in x y z of degree n and satisfies V xx + V yy + V zz = 0 is called a spherical harmonic. In polar coordinates, the spherical harmonics take the form r n f n , where f n is a spherical surface harmonic of degree n. On a sphere, f n satisfies f n + n n + 1 f n = 0, where is the spherical Laplacian. Bounded spherical surface harmonics are well studied, but in certain instances, unbounded spherical surface harmonics may be of interest. For example, if X is a parameterization of a minimal surface and n is the corresponding unit normal, it is known that the support function, w = X • n, satisfies w + 2w = 0 on a branched covering of a sphere with some points removed. While simple in form, the boundary value problem for the support function has a very rich solution set. We illustrate this by using spherical harmonics of degree one to construct a number of classical genus-zero minimal surfaces such as the catenoid, the helicoid, Enneper's surface, and Hennenberg's surface, and Riemann's family of singly periodic genus-one minimal surfaces.  2002 Elsevier Science (USA)