The generalized Cartan-Hadamard isoperimetric conjecture

Theorem 1.1: for $n\ge2$, $\kappa\le0$ and $(M^n,g)$ complete simply connected with $\mathrm{Sec}\le\kappa$, every measurable $E$ of finite volume and finite perimeter has $P(E)\ge\mathcal I_{\sqrt{-\kappa}}(\mathrm{vol}(E))$, the model-ball profile. Theorem 1.2: for $\mathrm{Sec}\le0$, a bounded positive-volume set attains the Euclidean bound iff it agrees a.e. with an open region whose induced metric is a round Euclidean ball (the ambient manifold need not be flat). Corollaries give Faber-Krahn and Saint-Venant comparisons with the model ball. Equality rigidity for $\kappa<0$ is not claimed. The companion proves the sharp Euclidean filling bound for integral $n$-cycles, $n\ge2$, in any proper CAT(0) space and deduces the Euclidean inequality for domains in dimension $d\ge3$. Recent preprints cited by the paper (Chen-Ghomi-Wang, dimensions 3 to 9) overlap in low dimensions.

Theorem 1.1: for $n\ge2$, $\kappa\le0$ and $(M^n,g)$ complete simply connected with $\mathrm{Sec}\le\kappa$, every measurable $E$ of finite volume and finite perimeter has $P(E)\ge\mathcal I_{\sqrt{-\kappa}}(\mathrm{vol}(E))$, the model-ball profile. Theorem 1.2: for $\mathrm{Sec}\le0$, a bounded positive-volume set attains the Euclidean bound iff it agrees a.e. with an open region whose induced metric is a round Euclidean ball (the ambient manifold need not be flat). Corollaries give Faber-Krahn and Saint-Venant comparisons with the model ball. Equality rigidity for $\kappa<0$ is not claimed. The companion proves the sharp Euclidean filling bound for integral $n$-cycles, $n\ge2$, in any proper CAT(0) space and deduces the Euclidean inequality for domains in dimension $d\ge3$. Recent preprints cited by the paper (Chen-Ghomi-Wang, dimensions 3 to 9) overlap in low dimensions.