A sharp quantitative estimate of critical sets

Abstract

The paper establishes a sharp quantitative estimate for the $(d-1)$-Hausdorff measure of the critical set of ${\mathcal{C}}^1$ vector-valued functions on ${\mathbb{R}}^d$. Additionally, we prove that for a generic ${\mathcal{C}}^2$ function where ``generic" is understood in the topological sense of Baire category, the critical set has a locally finite $(d-1)$-Hausdorff measure.

arXiv admin note: substantial text overlap with arXiv:2312.17462