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Zbl 0956.32028
García Barroso, Evelia; Teissier, Bernard
Multi-step concentration of the curvature in Milnor fibers. (Concentration multi-échelles de courbure dans des fibres de Milnor.) (French)
[J] Comment. Math. Helv. 74, No.3, 398-418 (1999). ISSN 0010-2571; ISSN 1420-8946/e

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. \par 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.
[T.de Jong (Saarbrücken)]

MSC 2000:: *32S55 Milnor fibration

Keywords: singularities; plane curves; Milnor fibration; curvature

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