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On Lipschitz Normally Embedded singularities

Lorenzo Fantini, Anne Pichon

TL;DR

This survey analyzes Lipschitz Normally Embedded (LNE) singularities by comparing inner $d_i$ and outer $d_o$ metrics on germs $(X,0)$ and exploring when the identity map is a bilipschitz equivalence between these metrics. It develops practical arc- and link-based criteria, including the pancake decomposition with the pancake metric $d_P$, and a finite-test-curve refinement for normal surface germs, to determine LNE in tractable cases. The discussion advances to complex surface germs, establishing results such as: (i) rational singularities are LNE iff minimal, (ii) superisolated hypersurfaces are LNE exactly when the tangent cone is reduced and LNE, and (iii) a broad new class of LNE hypersurfaces given by products minus a power, with a deeper analysis of polar curves and inner rates. The article also links LNE to tangent-cone properties, link topology, and polar geometry, and closes with open questions about behavior under blowups/Nash transforms, topological-type characterizations, and higher-dimensional generalizations, underscoring the significance of LNE as a unifying lens in the Lipschitz geometry of singularities.

Abstract

Any subanalytic germ $(X,0) \subset (\mathbb R^n,0)$ is equipped with two natural metrics: its outer metric, induced by the standard Euclidean metric of the ambient space, and its inner metric, which is defined by measuring the shortest length of paths on the germ $(X,0)$. The germs for which these two metrics are equivalent up to a bilipschitz homeomorphism, which are called Lipschitz Normally Embedded, have attracted a lot of interest in the last decade. In this survey we discuss many general facts about Lipschitz Normally Embedded singularities, before moving our focus to some recent developments on criteria, examples, and properties of Lipschitz Normally Embedded complex surfaces. We conclude the manuscript with a list of open questions which we believe to be worth of interest.

On Lipschitz Normally Embedded singularities

TL;DR

This survey analyzes Lipschitz Normally Embedded (LNE) singularities by comparing inner and outer metrics on germs and exploring when the identity map is a bilipschitz equivalence between these metrics. It develops practical arc- and link-based criteria, including the pancake decomposition with the pancake metric , and a finite-test-curve refinement for normal surface germs, to determine LNE in tractable cases. The discussion advances to complex surface germs, establishing results such as: (i) rational singularities are LNE iff minimal, (ii) superisolated hypersurfaces are LNE exactly when the tangent cone is reduced and LNE, and (iii) a broad new class of LNE hypersurfaces given by products minus a power, with a deeper analysis of polar curves and inner rates. The article also links LNE to tangent-cone properties, link topology, and polar geometry, and closes with open questions about behavior under blowups/Nash transforms, topological-type characterizations, and higher-dimensional generalizations, underscoring the significance of LNE as a unifying lens in the Lipschitz geometry of singularities.

Abstract

Any subanalytic germ is equipped with two natural metrics: its outer metric, induced by the standard Euclidean metric of the ambient space, and its inner metric, which is defined by measuring the shortest length of paths on the germ . The germs for which these two metrics are equivalent up to a bilipschitz homeomorphism, which are called Lipschitz Normally Embedded, have attracted a lot of interest in the last decade. In this survey we discuss many general facts about Lipschitz Normally Embedded singularities, before moving our focus to some recent developments on criteria, examples, and properties of Lipschitz Normally Embedded complex surfaces. We conclude the manuscript with a list of open questions which we believe to be worth of interest.
Paper Structure (18 sections, 18 theorems, 15 equations, 1 figure)

This paper contains 18 sections, 18 theorems, 15 equations, 1 figure.

Table of Contents

  1. Definition, first examples, and some general results
  2. Definition
  3. First examples
  4. The pancake decomposition, the pancake metric, and the embedding problem
  5. Characterization of LNE-ness via arcs
  6. Characterization of LNE-ness via the links
  7. LNE-ness and Moderately Discontinuous homology
  8. LNE-ness and tangent cones
  9. LNE in topology and other fields
  10. LNE among complex surface germs
  11. A refinement of the arc criterion
  12. Examples
  13. Properties of LNE surfaces
  14. Open questions
  15. Behavior under blowup and Nash transform
  16. ...and 3 more sections

Key Result

Theorem 1.9

Let $X \subset {\mathbb R}^n$ be a closed semialgebraic set. Then we can write as a finite union of closed semialgebraic subsets of such that:

Figures (1)

  • Figure 1: The real cusp $y^2-x^3=0$

Theorems & Definitions (39)

  • Definition 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Example 1.8
  • Theorem 1.9: Pancake Decomposition Parusinski1988Kurdyka1992Parusinski1994
  • Theorem 1.10: Pancake metric, KurdykaOrro1997
  • ...and 29 more