Anne Pichon

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.

Lipschitz geometry of complex curves

We describe the Lipschitz geometry of complex curves. To a large part this is well known material, but we give a stronger version even of known results. In particular, we give a quick proof, without any analytic restrictions, that the outer Lipschitz geometry of a germ of a complex plane curve determines and is determined by its embedded topology. This was first proved by Pham and Teissier, but in an analytic category. We also show the embedded topology of a plane curve determines its ambient Lipschitz geometry.

Milnor fibrations of meromorphic functions

In analogy with the holomorphic case, we compare the topology of Milnor fibrations associated to a meromorphic germ f/g: the local Milnor fibrations given on Milnor tubes over punctured discs around the critical values of f/g, and the Milnor fibration on a sphere.

COMPLEX ANALYTIC REALIZATION OF LINKS

We present the complex analytic and principal complex analytic realizability of a link in a 3-manifold

The boundary of the Milnor fiber of Hirzebruch surface singularities

M

Lipschitz Geometry Does not Determine Embedded Topological Type

as a tool for understanding the complex structures on the cone

The boundary of the Milnor fiber for some non-isolated singularities of complex surfaces

C(M)

Fibred Multilinks and singularities fg

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Real singularities and open-book decompositions of the 3-sphere

We give the first (as far as we know) complete description of the boundary of the Milnor fiber for some non-isolated singular germs of surfaces in

Lipschitz geometry of complex surfaces: analytic invariants and equisingularity

{\bf C}^3