Extendibility and boundedness of invariants on singularities of wavefronts

TL;DR

This paper explores the conditions under which geometric invariants like Gaussian and mean curvature remain bounded or extendable near singularities on wavefronts, linking these properties to uniform approximation by parallel surfaces.

Contribution

It provides necessary and sufficient conditions for the extendibility and boundedness of curvatures at singularities, and relates divergence of invariants to uniform approximation properties.

Findings

Conditions for boundedness of curvatures near singularities

Characterization of invariants' convergence to infinity

Relationship between invariants' behavior and surface approximation

Abstract

We investigate necessary and sufficient conditions for the extendibility and boundedness of Gaussian curvature, Mean curvature and principal curvatures near all types of singularities on fronts. We also study the convergence to infinite limits of these geometrical invariants and show how this is tightly related to a particular property of uniform approximation of fronts by parallel surfaces.