Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Donal O'Shea, Leslie Wilson

TL;DR

This work addresses the ambient bilipschitz classification of real surface germs in $\mathbb{R}^3$ with isolated singularities whose tangent cone is a plane, by analyzing the Nash and Zariski cones and the behavior of analytic arcs near the singularity. The authors develop a zone- and order-theory framework for arcs, introducing FI/FD/FL zones and a height function to capture how the tangent planes along arcs vary, and prove a key result (Theorem $6.6$) stating that, under mild hypotheses, ambient bilipschitz equivalence is governed by inner/outer metric equivalence (i.e., Lipschitz normal embedding). A central technical achievement is proving FL zones are closed via subanalytic preparation (Parusinski; van den Dries–Speissegger), enabling a constructive reduction to a flat model and establishing 1-regularity for the tested cases. The paper also provides explicit examples illustrating the intricate structure of Nash cones in the real setting and discusses potential extensions to more general tangent cones, outlining a path toward a broader ambient bilipschitz classification of real surface singularities.

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.

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

TL;DR

This work addresses the ambient bilipschitz classification of real surface germs in with isolated singularities whose tangent cone is a plane, by analyzing the Nash and Zariski cones and the behavior of analytic arcs near the singularity. The authors develop a zone- and order-theory framework for arcs, introducing FI/FD/FL zones and a height function to capture how the tangent planes along arcs vary, and prove a key result (Theorem ) stating that, under mild hypotheses, ambient bilipschitz equivalence is governed by inner/outer metric equivalence (i.e., Lipschitz normal embedding). A central technical achievement is proving FL zones are closed via subanalytic preparation (Parusinski; van den Dries–Speissegger), enabling a constructive reduction to a flat model and establishing 1-regularity for the tested cases. The paper also provides explicit examples illustrating the intricate structure of Nash cones in the real setting and discusses potential extensions to more general tangent cones, outlining a path toward a broader ambient bilipschitz classification of real surface singularities.

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 with isolated singularity and whose tangent cone is a plane.
Paper Structure (9 sections, 14 theorems, 88 equations)

This paper contains 9 sections, 14 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Zariski and Nash cones
  3. Bilipschitz geometry
  4. The case when the tangent cone is a plane
  5. Connected families of analytic arcs in $\mathbb R^2$
  6. Increasing, decreasing, and flat zones and their heights
  7. Examples
  8. Proof that $FL$ zones are closed
  9. Extending Theorem 6.6

Key Result

Lemma 5.1

Suppose that $A$ is a zone (hence, connected) with $w(A) = w$. Then there is a unique finite Puiseux series with $r_i$ positive rational numbers $r_1 < \ldots < r_n < \frac{1}{w}$ such that each $\alpha \in A$ has the form ($h.o.t._{\ge \frac{1}{w}}$ denotes a sum of non-zero terms of order greater than or equal to $\frac{1}{w}$).

Theorems & Definitions (26)

  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • Definition 5.4
  • Lemma 5.5
  • proof
  • Lemma 5.6
  • proof
  • ...and 16 more