Gauss–Kronecker curvature and equisingularity at infinity of definable families

Abstract

Assume given a polynomially bounded o-minimal structure expanding the real numbers. Let (Ts) s∈R be a definable family of C 2 -hypersurfaces of R n . Upon defining the notion of generalized critical value for such a family, we show that the functions s → |K|(s) and s → K(s), respectively the total absolute Gauss-Kronecker and total Gauss-Kronecker curvature of Ts, are continuous in any neighbourhood of any value which is not generalized critical. In particular this provides a necessary criterion of equisingularity for the family of the levels of a real polynomial.

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

“…In this last section we expand and strengthen the study initiated in [17], addressing here the special case of families of hypersurfaces.…”

“…n−1 , and of our recent result [17], where continuity of GK and |GK| is proved at any regular value at which the function is also spherically regular at infinity. It is worth mentioning here the recent paper [11] investigating sufficient conditions of equi-singularity at infinity in terms of topological properties of set valued mappings over the levels of the given mapping F .…”

“…There are many occurrences of this result over the last forty years, mostly for functions though [6,10,17,22,23,24,26,32,37,39,40,46,47] (this list is far from being exhaustive). The proof we present here is explicit and does not get into the messy task of composing s flows (which is rarely explicitly done once s ⩾ 2).…”

“…In other words we have proved that In other words, we have rigorously showed, in this most general context, that (MR)-regularity and (the current avatar of) t-regularity are equivalent (see [10,17,46] for special cases).…”

“…When X and P are affine spaces, sufficient conditions for F or W to be locally trivial at a regular value already exist. Among others are Malgrange-Rabier condition [26,38,39], t-regularity [10,40,45,46], ρ-regularity [10,32,34], spherical-ness at infinity [7,8,17]. Any such regularity condition at infinity at a given value compels the behaviour at infinity of the nearby levels to be "tame".…”