Special functions on the sphere with applications to minimal surfaces

  • Authors:

  • Frank Baginski

  • Affiliations:

  • Department of Mathematics, The George Washington Universityy, Washington, DC

  • Venue:
  • Advances in Applied Mathematics - Special issue: Memory of Rodica Simon
  • Year:
  • 2002

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Abstract

A function which is homogeneous in x, y, z of degree n and satisfies Vxx + Vyy + Vzz = 0 is called a spherical harmonic. In polar coordinates, the spherical harmonics take the form rnfn, where fn, is a spherical surface harmonic of degree n. On a sphere, fn satisfies Δfn + n(n + 1)fn= 0, where Δ 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 Δw + 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.