A sharp quantitative estimate of critical sets

TL;DR

This paper provides precise estimates for the size of the critical set of smooth functions and shows that generically, this set has finite measure, advancing understanding in geometric analysis.

Contribution

It offers a sharp quantitative estimate for the Hausdorff measure of critical sets and proves generic finiteness for smooth functions.

Findings

Established a sharp estimate for the Hausdorff measure of critical sets.

Proved that generically, the critical set has locally finite measure.

Contributed to the geometric understanding of critical sets in smooth functions.

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.