Stratified critical points one the real Milnor fibre and integral-geometric formulas

TL;DR

This paper investigates the critical points of functions on real Milnor fibres of subanalytic sets, establishing relations to topology and deriving integral-geometric formulas, including an asymptotic Gauss-Bonnet theorem.

Contribution

It introduces a novel connection between critical points and topology of Milnor fibres, and derives new integral-geometric formulas for subanalytic sets.

Findings

Relation between critical points and topology of Milnor fibres

Asymptotic Gauss-Bonnet formula for real Milnor fibres

Infinitesimal linear kinematic formulas

Abstract

Let $(X,0) \subset (\mathbb{R}^n,0)$ be the germ of a closed subanalytic set and let $f$ and $g : (X,0) \rightarrow (\mathbb{R},0)$ be two subanalytic functions. Under some conditions, we relate the critical points of $g$ on the real Milnor fibre $X \cap f^{-1}(\delta) \cap B_\epsilon$, $0 <| \delta | \ll \epsilon \ll 1$, to the topology of this fibre and other related subanalytic sets. As an application, when $g$ is a generic linear function, we obtain an "asymptotic" Gauss-Bonnet formula for the real Milnor fibre of $f$. From this Gauss-Bonnet formula, we deduce "infinitesimal" linear kinematic formulas.