Equi-singularity of real families and Lipschitz Killing curvature densities at infinity

Abstract

Fix an o-minimal structure expanding the ordered field of real numbers. Let $(W_y)_{y\in\mathbb{R}^s}$ be a definable family of closed subsets of $\mathbb{R}^n$ whose total space $W = \cup_y W_y\times y$ is a closed connected $C^2$ definable sub-manifold of $\mathbb{R}^n\times\mathbb{R}^s$. Let $\varphi:W \to\mathbb{R}^s$ be the restriction of the projection to the second factor. After defining $K(\varphi)$, the set of generalized critical values of $\varphi$, showing that they are closed and definable of positive codimension in $\mathbb{R}^s$, contain the bifurcation values of $\varphi$ and are stable under generic plane sections, we prove that all the Lipschitz-Killing curvature densities at infinity $y \mapsto \kappa_i^\infty(W_y)$ are continuous functions over $\mathbb{R}^s\setminus K(\varphi)$. When $W$ is a $C^2$ definable hypersurface of $\mathbb{R}^n\times\mathbb{R}^s$, we further obtain that the symmetric principal curvature densities at infinity $y \mapsto \sigma_i^\infty(W_y)$ are continuous functions over $\mathbb{R}^s\setminus K(\varphi)$.