Isometric deformations of cuspidal edges

TL;DR

This paper studies isometric deformations of cuspidal edges on surfaces in 3D, showing that generic cuspidal edges can be deformed while preserving certain curvature properties into planar singularities, with uniqueness results.

Contribution

It introduces a method to deform generic cuspidal edges isometrically while maintaining the limiting normal curvature, leading to planar singularities with unique determination.

Findings

Generic cuspidal edges can be deformed into planar singularities.

The limiting normal curvature is preserved during deformation.

Deformed cuspidal edges are uniquely determined by initial data.

Abstract

Along cuspidal edge singularities on a given surface in Euclidean 3-space, which can be parametrized by a regular space curve, a unit normal vector field $\nu$ is well-defined as a smooth vector field of the surface. A cuspidal edge singular point is called generic if the osculating plane of the cuspidal edge (as a regular space curve) is not orthogonal to $\nu$. This genericity is equivalent to the condition that its limiting normal curvature $\kappa_\nu$ takes a non-zero value. In this paper, we show that a given generic (real analytic) cuspidal edge can be isometrically deformed preserving $\kappa_\nu$ into a cuspidal edge whose singular set lies in a plane. Such a limiting cuspidal edge is uniquely determined from the initial germ of the cuspidal edge.