Curvature integrals on the real Milnor fibre

  • Nicolas Dutertre

    Université de Provence, Marseille, France


Let f : ℝn + 1 → ℝ be a polynomial with an isolated critical point at 0 and let ft : ℝn + 1 → ℝ be a one-parameter deformation of f. We study the differential geometry of the real Milnor fiber Ctε = ft−1 (0) ∩ Bεn + 1. More precisely, we express the limits

limε → 0 limt → 0 1 ⁄εk ∫Ctε sn − k (x) dx,

where sn − k is the (n − k)-th symmetric function of curvature, in terms of the following averages of topological degrees:

∫Gn+1k deg0 ∇(f |H) dH,

where Gn+1k is the Grassmann manifold of k-dimensional planes through the origin of ℝn + 1. When 0 is an algebraically isolated critical point, we study the limits

limε → 0 limt → 0 1 ⁄εk ∫Ctε hn − k (x) dx,

where the hn − k are positive extrinsic curvature functions. We prove that these limits are finite and that they are bounded in terms of the Milnor–Teissier numbers of the complexification of f.

Cite this article

Nicolas Dutertre, Curvature integrals on the real Milnor fibre. Comment. Math. Helv. 83 (2008), no. 2, pp. 241–288

DOI 10.4171/CMH/124