Isoperimetric inequalities on manifolds with curvature bounds

The celebrated Euclidean isoperimetric inequality provides a sharp estimate of the measure of the boundary of subsets of the Euclidean space in terms of their volume. This inequality is also rigid, as equality is achieved only by Euclidean balls. The study of sharp and rigid isoperimetric inequalities on Riemannian manifolds has been an active area of research over the past few decades and is closely connected to curvature bounds.
In this introductory talk, we will review some classical and recent isoperimetric inequalities on classes of Riemannian manifolds with curvature bounded below. A possible unified approach to the proofs of such inequalities arises from the sharp concavity properties of the isoperimetric profile function. This latter result was first obtained on compact manifolds by C. Bavard--P. Pansu and S. Gallot in the '80s, and it has recently been extended to the noncompact setting in a joint work with G. Antonelli, E. Pasqualetto, and D. Semola.