Croke–Dairbekov–Sharafutdinov filling-volume rigidity conjecture

Let $X$ be a compact subdomain of a simply connected Riemannian manifold of dimension at least $3$ with negative, or possibly nonpositive, curvature, and let $\partial X$ carry the chordal metric induced by the ambient domain. The filling volume is computed using fillings whose boundary distance dominates this chordal metric.

Croke–Dairbekov–Sharafutdinov filling-volume conjecture. The filling volume of $\partial X$ with its chordal metric equals $\operatorname{vol}(X)$, and $X$ is the unique volume-minimizing filling up to isometry.

Known cases include Euclidean and certain negatively curved symmetric-space domains, while the general negatively curved case remains open. The parenthetical uncertainty in the source leaves the curvature hypothesis ambiguous.