Deformations of swallowtails in a 3-dimensional space form

TL;DR

This paper provides a representation formula for swallowtail singularities in Euclidean 3-space, investigates their properties, and shows how they deform while preserving the sign of their Gaussian curvature.

Contribution

It introduces a new representation formula for swallowtails and analyzes their deformation behavior in 3D space, extending previous work on cuspidal edges.

Findings

Representation formula for swallowtails in Euclidean 3-space.

Swallowtails can be deformed into ones with constant Gaussian curvature.

Deformations preserve the sign of Gaussian curvature.

Abstract

This is a continuation of the authors' earlier work on deformations of cuspidal edges. We give a representation formula for swallowtails in the Euclidean 3-space. Using this, we investigate map germs of generic swallowtails in 3-dimensional space from, and show some important properties of them. In particular, we give a representation formula giving all map germs of swallowtails in the Euclidean 3-space whose Gaussian curvatures are bounded from below by a positive constant or by a negative constant from above. Using this, we show that any swallowtails are deformed into a swallowtail of constant Gaussian curvature preserving the sign of their Gaussian curvatures.