Quantitative stability of conformal parameterization under mesh perturbations

Derive a quantitative stability estimate for the centered conformal parameterization of triangulated surfaces in terms of mesh perturbations, including the dependence on the conformal-barycenter Hessian and the relevant mesh or geometric norms.

Background

The paper establishes stability of the polynomial invariant map with respect to orbit distance between coefficient descriptors. It also states that uniformizing maps depend continuously on the metric and that the conformal barycenter is smooth on stable measures, with derivative control related to the Hessian of the barycenter energy.

A quantitative estimate connecting perturbations of a triangulated mesh to changes in the discrete conformal parameterization and the resulting centered coefficients is not supplied. Such an estimate would complement the coefficient-to-invariant stability theorem and provide a rigorous basis for mesh-level error control.