Lev Birbrair

The thick-thin decomposition and the bilipschitz classification of normal surface singularities

We describe a natural decomposition of a normal complex surface singularity (X, 0) into its “thick” and “thin” parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts.By refining the thick-thin decomposition, we then give a complete description of the intrinsic bilipschitz geometry of (X, 0) in terms of its topology and a finite list of numerical bilipschitz invariants.

BI-LIPSCHITZ GEOMETRY OF WEIGHTED HOMOGENEOUS SURFACE SINGULARITIES

We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz equivalent.

Bi-Lipschitz geometry of complex surface singularities

We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface with particular emphasis on when it is metrically conical.

-bi-Lipschitz equivalence of real function-germs

In this paper we prove that the set of equivalence classes of germs of real polynomials of degree less than or equal to k, with respect to κ-bi-Lipschitz equivalence, is finite.

Separating sets, metric tangent cone and applications for complex algebraic germs

An explanation is given for the initially surprising ubiquity of separating sets in normal complex surface germs. It is shown that they are quite common in higher dimensions too. The relationship between separating sets and the geometry of the metric tangent cone of Bernig and Lytchak is described. Moreover, separating sets are used to show that the inner Lipschitz type need not be constant in a family of normal complex surface germs of constant topology.

Arc Criterion of Normal Embedding

We present a criterion of local normal embedding of a semialgebraic (or definable in a polynomially bounded o-minimal structure) germ contained in \(\mathbb R^n\) in terms of orders of contact of arcs. Namely, we prove that a semialgebraic germ is normally embedded if and only if for any pair of arcs, coming to this point the inner order of contact is equal to the outer order of contact.

Real and Complex Singularities: On normal embedding of complex algebraic surfaces

We construct examples of complex algebraic surfaces not admitting normal embeddings (in the sense of semialgebraic or subanalytic sets) with image a complex algebraic surface.

Medial Axis and Singularities

This paper is devoted to the study of the medial axes of sets definable in polynomially bounded o-minimal structures, i.e. the sets of points with more than one closest point with respect to the Euclidean distance. Our point of view is that of singularity theory. While trying to make the paper self-contained, we gather here also a large bunch of basic results. Our main interest, however, goes to the characterization of those singular points of a definable, closed set

Metrically un-knotted corank 1 singularities of surfaces in

X\subset {\mathbb {R}}^n