Inspired by the classical Poincaré and Korn inequalities in analysis, the "Poincaré-Korn" inequalities were introduced in 2020 by Carrapatoso et al. In that paper, the authors conjectured that among a suitable class of reference measures, the standard Gaussian measure uniquely achieves the smallest sharp constant in the Poincaré-Korn inequality. In this talk, I'll establish a quantitative stability result for the Poincaré-Korn inequalities, affirmatively resolving the question by Carrapatoso et al. The argument involves a combination of Stein's method and variational techniques. Time permitting, I'll also mention recent work on stability of Klartag's improved Lichnerowicz bound, which is one of the main ingredients behind current best-progress on the KLS conjecture. Everything is joint work with Max Fathi.
