Multi-step concentration of the curvature in Milnor fibers
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.
scientific article; zbMATH DE number 1375329