Remark: Family 337. Cartan–Hadamard isoperimetry and CAT (0) fillings

Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most $\kappa \le 0$, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for $\kappa =0$ are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral $n$-cycles, $n\ge 2$, in arbitrary proper CAT$(0)$ spaces.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.