Equivalence of real Milnor's fibrations for quasi homogeneous singularities

TL;DR

This paper demonstrates the equivalence of different Milnor fibrations for real quasi-homogeneous singularities with isolated critical points, using Euler vector fields to explicitly characterize critical points.

Contribution

It introduces a method to establish the equivalence of Milnor fibrations in different settings for real quasi-homogeneous singularities, expanding understanding of their topological structure.

Findings

Milnor fibrations in a hollow tube and in the complement of the link are equivalent.

Explicit characterization of critical points of the projection map.

Use of Euler vector fields to analyze singularity fibrations.

Abstract

We are going to use the Euler's vector fields in order to show that for real quasi-homogeneous singularities with isolated critical value, the Milnor's fibration in a "thin" hollowed tube involving the zero level and the fibration in the complement of "link" in sphere are equivalents, since they exist. Moreover, in order to do that, we explicitly characterize the critical points of projection $\frac{f}{\|f\|}:S_{\epsilon}^{m}\setminus K_{\epsilon}\to S^{1}$, where $K_{\epsilon}$ is the link of singularity.