Isoperimetric Inequalities and Upper Curvature Bounds

The isoperimetric function, or Dehn function, measures the area required to fill a closed curve of a given length by a disc. In this talk, I discuss the relationship between upper curvature bounds in the sense of Alexandrov and isoperimetric inequalities. In particular, I will outline the proof of the following characterization: a geodesic metric space has curvature bounded above by a constant $(\kappa)$ if and only if its Dehn function is bounded above by the Dehn function of the simply connected surface of constant curvature $(\kappa)$. In the locally compact setting, this result was established in joint work with Alexander Lytchak, while the general case was proved more recently together with Stephan Stadler. I will also discuss recent joint work with Toni Ikonen, where we prove a stability theorem for Dehn functions under ultralimits of metric spaces. This yields a substantially simpler proof of the general characterization theorem.