Geometry of bifurcation sets of generic unfoldings of corank two functions

TL;DR

This paper explores the geometric structure of bifurcation sets in generic unfoldings of corank two functions, providing parametrizations and analyzing curvature and special curves near singular points.

Contribution

It introduces a parametrization of bifurcation sets as surfaces in three-dimensional space and studies their curvature and special curves, advancing understanding of their geometric properties.

Findings

Bifurcation sets admit a surface parametrization in R^3.

Gaussian and principal curvatures are analyzed near singular points.

Number of ridge and subparabolic curves near singularities is determined.

Abstract

We study the geometry of bifurcation sets of generic unfoldings of $D_4^\pm$-functions. Taking blow-ups, we show each of the bifurcation sets of $D_4^\pm$-functions admit a parametrization as a surface in $R^3$. Using this parametrization, we investigate the behavior of the Gaussian curvature and the principal curvatures. Furthermore, we investigate the number of ridge curves and subparabolic curves near their singular point.