Bilipschitz geometry of real surface singularities whose tangent cone is a plane

TL;DR

This paper investigates how the Nash cone and exceptional rays of real surface singularities with a planar tangent cone behave under ambient bilipschitz transformations, revealing nuanced geometric invariants.

Contribution

It provides new insights into the invariance and behavior of the Nash cone and exceptional rays for real surface singularities with planar tangent cones under bilipschitz equivalence.

Findings

Tangent cones are preserved under ambient bilipschitz equivalence.

The Nash cone's behavior is more delicate and not necessarily preserved.

Exceptional rays exhibit specific transformation properties under bilipschitz maps.

Abstract

Tangent cones are preserved under ambient bilipschitz equivalence, but the behavior of the Nash cone is more delicate. This paper explores the behavior of the Nash cone and of exceptional rays under ambient bilipschitz equivalence for real surfaces in $\mathbb R^3$ with isolated singularity and whose tangent cone is a plane.