Sard's theorem

Abstract: of a region R of euclidean m-space into part of euclidean w-space. Suppose that each f unction ƒ' 0' = 1, • • • , n) is of class C in R (q^l). A critical point of the map (1.1) is a point in R at which the matrix of first derivatives 2)? = ||/*|| (i = ly • • • , m;j = l, • • • , n) is of less than maximum rank. The rank of a critical point # is the rank of 5DÎ at x. A critical value is the image under (1.1) of a critical point. If » = 1, these definitions are the usual definitions of critical point and value of a continuously differentiable function. We prove the following result: If m^n, the set of critical values of the map (1.1) is of m-dimensional measure zero without further hypothesis on q; if m>n, the set of critical values of the map (1.1) is of n-dimensional measure zero providing that q^m — n + 1. Using an example due to Hassler Whitney we show that the hypothesis on q cannot be weakened. We prove also that the critical values of (1.1) corresponding to critical points of rank zero constitute a set of (m/q)dimensional measure zero. The idea of considering the measure of the set of critical values of one function or of several functions is due to Marston Morse. The first result stated above reduces, if » = 1, to the known theorem : The critical values of a function of m variables of class C constitute a set of linear measure zero. A. P. Morse has given a proof of this theorem for all m. In the present paper we make use of one of A. P. Morse's results.

Abstract: We prove that a semialgebraic differentiable mapping has a generalized critical values set of measure zero. Moreover, if the mapping is C2 we obtain, by a generalisation of Ehresmann's fibration theorem due to P. J. Rabier [20], a locally trivial fibration over the complement of this set. In the complex case, we prove that the set of generalized critical values of a polynomial mapping is a proper algebraic set.

Abstract: We prove various generalizations of classical Sard's theorem to mappings f:Mm→Nn between manifolds in Holder and Sobolev classes. It turns out that if f ∈ Ck,λ(Mm,Nn), then—for arbitrary k and λ—one can obtain estimates of the Hausdorff measure of the set of critical points in a typical level set f−1(y). The classical theorem of Sard holds true for f ∈ Ck with sufficiently large k, i.e., k>max(m−n,0); our estimates contain Sard's theorem (and improvements due to Dubovitskii and Bates) as special cases. For Sobolev mappings between manifolds, we describe the structure of f−1(y).