Harmonic Analysis on Manifolds: Cut Loci, LaplaceBeltrami Operators, and Computational Methods

The author addresses the problem of solving the Poisson and LaplaceBeltrami equations on manifolds with complex topology. Classical approaches such as Fourier and spectral methods provide exact or near-exact solutions for manifolds with simple global parameterizations, but they fail in the presence of complex topologies or in the absence of highly symmetrical structure. Computational methods, while effective, are often limited by error tolerance and high computational complexity. To overcome these limitations, A distance-parameterization framework that constructs cuts on the manifold using geodesic wavefront propagation is proposed. This ensures that the resulting manifold is simply connected, without holes, allowing the use of analytic methods under well-defined boundary conditions. By using Fourier analysis, the author substituted the equation to a convolution-like form, where solutions for Minkowski metric can be easily obtained. However, it is hard to obtain an analytic solution for arbitrary metric from the equation, where the author must turn back to numerical solutions for integration.