A Sharp Isoperimetric Gap Theorem in Non-Positive Curvature

This lecture explores isoperimetric inequalities for null-homotopies of Lipschitz 2-spheres in proper CAT(0) spaces, presenting joint research by Urs Lang, Cornelia Drutu, Panos Papasoglu, and Stephan Stadler. Discover how their work establishes a significant gap theorem in higher dimensions that yields exponents approaching 1, analogous to known results in one dimension less where quadratic inequalities below the sharp threshold of 1/(4π) imply Gromov hyperbolicity and linear inequalities. Learn about their proof of a previously undocumented Euclidean isoperimetric inequality for null-homotopies of 2-spheres and the introduction of minimal tetrahedra that satisfy a linear inequality.