Generalized Cartan--Hadamard isoperimetry and Euclidean equality rigidity

We resolve the generalized Cartan--Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most $\kappa\le0$, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature $\kappa$. For bounded positive-volume sets, equality in the Euclidean comparison holds precisely when the set agrees up to null sets with an open region isometric, with its induced metric, to a round Euclidean ball.