Multi-step concentration of the curvature in Milnor fibers
type: Item
instance of: scholarly article
title: Multi-step concentration of the curvature in Milnor fibers
zbMATH Open document ID: 0956.32028
author: Bernard Teissier
author: García Evelia R. Barroso
published in: Commentarii Mathematici Helvetici
publication date: 2000-05-29T00:00:00Z
review text: Let be given a germ \(f\) of a holomorphic function in two variables, defining a germ of a complex curve \(C\). Let \(K\) be the Lipschitz-Killing curvature on the Milnor fibre \(C(\lambda)_\varepsilon = f^{-1}(\lambda) \cap B_\varepsilon\) with \(|\lambda|\ll \varepsilon\). Langevin showed that the integral of \(|K|\) tends to \(2\pi(\mu(C) + m(C)-1)\), if \(\varepsilon\) tends to zero, where \(\mu(C)\) is the Milnor number and \(m(C)\) is the multiplicity. In this paper the authors show that a large part of the curvature is asymptotically contained in balls, whose centers can be described, and whose radii are of type \(|\lambda|^{\rho}\), where the \(\rho\)'s are rational numbers that depend only on the topological type of \(C\). In case that \(C\) is irreducible (or more generally, there is just one tangential direction), they in fact show that all of the curvature is asymptotically totally contained in these balls.
Mathematics Subject Classification ID: 32S55
zbMATH DE Number: 1375329
zbMATH Keywords: singularities
zbMATH Keywords: plane curves
zbMATH Keywords: Milnor fibration
zbMATH Keywords: curvature
MaRDI profile type: Publication
full work available at URL: s000140050096
OpenAlex ID: W1986837346
DOI: 10.1007/S000140050096
description: scientific article; zbMATH DE number 1375329