Commentarii Mathematici Helvetici

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Volume 83, Issue 2, 2008, pp. 241–288
DOI: 10.4171/CMH/124

Published online: 2008-06-30

Curvature integrals on the real Milnor fibre

Nicolas Dutertre[1]

(1) 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.

Keywords: Real Milnor fibre, curvatures, topological degree

Dutertre Nicolas: Curvature integrals on the real Milnor fibre. Comment. Math. Helv. 83 (2008), 241-288. doi: 10.4171/CMH/124