Conformal Mapping with as Uniform as Possible Conformal Factor

Abstract

According to the uniformization theorem, any surface can be conformally mapped into a domain of a constant Gaussian curvature. The conformal factor indicates the local scaling introduced by such a mapping. This process could be used to compute geometric quantities in a simplified flat domain with zero Gaussian curvature. For example, the computation of geodesic distances on a curved surface can be mapped into solving an eikonal equation in a plane weighted by the conformal factor. Solving an eikonal equation on the weighted plane can then be done by regular sampling of the domain using, for example, the celebrated fast marching method (FMM). The connection between the conformal factor on the plane and the surface geometry can be justified analytically. Still, in order to construct consistent numerical solvers that exploit this relation, one needs to prove that the conformal factor is bounded. We provide theoretical bounds of the conformal factor and introduce optimization formulations that control its beh...