Celebratio Mathematica

W. D. Neumann and A. Pichon: Lipschitz geometry of complex surfaces: Analytic invariants and equisingularity. Technical report, November 2012. ArXiv 1211.4897 techreport

L. Birbrair, W. D. Neumann, and A. Pichon: “The thick-thin decomposition and the bilipschitz classification of normal surface singularities,” Acta Math. 212 : 2 (2014), pp. 199–256. MR 3207758 Zbl 1303.14016 ArXiv 1105.3327 article

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.

Anne Pichon	Related
Lev B. Birbrair	Related

W. D. Neumann and A. Pichon: “Lipschitz geometry of complex curves,” J. Singul. 10 (2014), pp. 225–234. MR 3300297 Zbl 1323.14003 ArXiv 1302.1138 article

W. D. Neumann, H. M. Pedersen, and A. Pichon: “A characterization of Lipschitz normally embedded surface singularities,” J. Lond. Math. Soc. (2) 101 : 2 (2020), pp. 612–640. MR 4093968 Zbl 1441.14015 ArXiv 1806.11240 article

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. These two metrics are in general nonequivalent up to bilipschitz homeomorphism. We give a necessary and sufficient condition for a normal surface singularity to be Lipschitz normally embedded (LNE), that is, to have bilipschitz equivalent outer and inner metrics. In a partner paper (Neumann, Pedersen and Pichon, J. London Math. Soc. (2020)), we apply it to prove that rational surface singularities are LNE if and only if they are minimal.

Anne Pichon	Related
Helge Møller Pedersen	Related

W. D. Neumann, H. M. Pedersen, and A. Pichon: “Minimal surface singularities are Lipschitz normally embedded,” J. Lond. Math. Soc. (2) 101 : 2 (2020), pp. 641–658. MR 4093969 Zbl 1441.14016 ArXiv 1503.03301 article

Anne Pichon	Related
Helge Møller Pedersen	Related

Introduction to Lipschitz geometry of singularities (Cuernavaca, Mexico, 11–22 June 2018). Edited by W. Neumann and A. Pichon. Lecture Notes in Mathematics 2280. Springer (Cham, Switzerland), 2020. MR 4200092 Zbl 1456.58002 book

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Walter D. Neumann

Contributions on Lipschitz geometry of complex singularities

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