Cartan–Hadamard conjecture

A circle encloses a given flat area with the least perimeter; in three dimensions the corresponding benchmark is a ball. Can a space that curves flat or negatively in every two-dimensional direction ever enclose the same volume with less boundary than a flat-space ball? The conjecture says no, in every dimension. The space must be complete and have no loops that are trapped by topological holes; bounded regions may have nonsmooth boundaries of finite perimeter.[1] [2]

Among regions enclosing the same volume, the Euclidean ball has the smallest boundary. Does that benchmark remain a lower bound when the ambient space has nonpositive sectional curvature and no topological holes? For a flat circle of radius $r$, $P=2\pi r$ and $A=\pi r^2$, so $P^2=4\pi A$; this is the two-dimensional benchmark, not a proof in higher dimensions.

For every complete simply connected Riemannian $M^n$ with $\mathrm{sec}\le0$ and $n\ge2$, must every bounded positive-volume region $\Omega$ of finite perimeter satisfy $P(\Omega)^n\ge n^n\omega_n\operatorname{Vol}(\Omega)^{n-1}$? Here $\omega_n$ is the Euclidean unit-ball volume.[1] [2]

Known low-dimensional results and a recent preprint claiming dimensions three through nine do not cover every dimension. A separate preprint handles sufficiently small volumes with an extra lower Ricci-curvature bound. The missing comparison must work for all allowed volumes and dimensions without those added restrictions.[1] [2]

The n=3–9 result is a recent preprint claim. Small-volume estimates with additional Ricci control do not prove all volumes in arbitrary dimension.[1] [2]

The flat ball supplies a concrete efficiency benchmark for enclosure. Comparing curved spaces with that benchmark tests whether a local curvature condition can prevent a global saving in boundary area. It connects how geometry bends in small directions to the behaviour of whole volume-enclosing regions.[1] [2]

The question turns local curvature control into a sharp global variational constraint. Boundary minimizers are linked to constant-mean-curvature geometry, and the recent dimension-restricted work makes that link explicit. A sharp comparison, together with an appropriate equality-rigidity statement, helps distinguish genuinely flat minimizing regions from curved alternatives; the unrestricted conjecture cannot be inferred from a restricted equality theorem.[1]