Conformal Spherical Splines for the Laplace-Beltrami Operator on Genus-Zero Surfaces

This paper addresses the numerical solution of partial differential equations on curved surfaces. It develops a method using smooth spherical splines to solve the Poisson equation and eigenvalue problems for the Laplace-Beltrami operator on genus-zero surfaces. The approach represents surface geometry solely through a conformal factor applied to the round metric, leveraging the conformal invariance of Dirichlet energy in two dimensions. The work provides algorithmic construction details and experimental validation for this computational geometry approach.