In case a sharp functional inequality admits optimizers, we are interested in improving the inequality by adding terms that involve a distance to the set of optimizers. Such refinements are known as (quantitative) stability results.

In this talk, I first provide a short introduction to the topic of stability of functional inequalities. Following this, I present the σ2-curvature inequality, a variational characterization of a fully nonlinear Yamabe-type equation, and explain how stability of this inequality can be established. As we will see, in contrast to previous Hilbert-space results, the distance to the set of optimizers is measured naturally in terms of two different Sobolev norms, for which optimal exponents are provided. Finally, I describe how the presented methods can be applied to improve an inequality by De Lellis and Topping, which in turn is a refinement of a well-known rigidity result by Schur. The talk is based on two joint works with Rupert Frank and Tobias König, respectively.