Curvature integrals on the real Milnor fibre

Abstract

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.