ON THE CURVATURE OF THE REAL MILNOR FIBER

Abstract

Let $C$ be a germ at $O \in {\mathbb R}^2$ of a real analytic plane curve, and $C^{\mathbb C}$ its complexification; let $C_t \subset B_{\varepsilon}$ be a fiber of a real smooth deformation of $C$ in the ball $B_{\varepsilon} = B(O, \varepsilon)$. The following inequality is proved between the integrals of real curvature $k$ of $C_t$ and those of Gaussian curvature $K$ of $C_{t}^{\mathbb C}$: $$ 2 \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^\varepsilon} \vert k \vert \leq \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^{\varepsilon {\mathbb C}}} \vert K \vert.$$ The sharpness of this inequality is proved in the case where $C$ is a real irreducible germ. Similar results are proved for an affine algebraic curve $C \subset {\mathbb R}^2$ of degree $d$.