1 · The essentials
How little boundary can enclose a given amount of space? These manuscripts study that question in spaces whose curvature is zero or negative, under specific conditions on their geometry. The main manuscript states that a region cannot have less boundary than a ball enclosing the same amount of space in the corresponding constant-curvature setting. Its comparison applies in every dimension, provided the region has finite volume and finite boundary size. For the zero-curvature comparison, a bounded region of positive volume reaches equality exactly when it is a round Euclidean ball in its own geometry, apart from differences of zero volume. A companion manuscript gives a related sharp bound on how much material is needed to fill closed objects in a broader class of spaces. The main result has a Lean formalization, although the supplied material does not specify exactly which accompanying claims it covers.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine enclosing a fixed amount of land with fencing: the question is how little fence the geometry permits. Here the benchmark is a ball in a matching constant-curvature space, but the result concerns higher-dimensional regions as well as ordinary plots of land.
Why it matters
The main manuscript describes its result as resolving the generalized Cartan–Hadamard isoperimetric conjecture in every dimension. It also identifies exactly which bounded positive-volume regions attain the Euclidean benchmark, rather than giving only an inequality. The companion result extends sharp Euclidean filling bounds to proper CAT(0) spaces, with arbitrary integer multiplicities and no restriction on ambient dimension.
2 · Going further
The precise comparison concerns a complete, simply connected smooth manifold whose sectional curvature is at most . For every set of finite volume and finite ambient perimeter, the main manuscript states that its perimeter is at least the perimeter of an equal-volume ball in the constant-curvature model. The abstract describes this as a resolution of the generalized Cartan–Hadamard isoperimetric conjecture in every dimension.
For the Euclidean comparison, , the manuscript also gives an equality characterization. A bounded set of positive volume attains equality precisely when it agrees, up to a set of zero volume, with an open region that is isometric with its induced metric to a round Euclidean ball. This is a statement about the equality region; it does not assert that the whole surrounding manifold is Euclidean.
The companion manuscript treats compactly supported integral -cycles, with , in arbitrary proper CAT(0) spaces. It states that each such cycle bounds a compactly supported integral current satisfying the sharp Euclidean mass bound, allowing arbitrary integer multiplicities and unrestricted ambient dimension. Its abstract identifies Euclidean Cartan–Hadamard isoperimetry in dimensions at least three as a consequence. The supplied abstracts state these results but do not describe the argument, so they do not support a more specific account of the proof strategy or the dependence between the two manuscripts.
Useful notions
- Volume and boundary size
- Curvature and constant-curvature spaces
- Completeness and simple connectivity
- Equality cases in inequalities
- Cycles, currents, and fillings
What was known before / what is new
Before
The main abstract identifies the generalized Cartan–Hadamard isoperimetric statement as a conjecture. The companion abstract likewise refers to the Euclidean Cartan–Hadamard isoperimetric conjecture. The supplied material gives no account of earlier partial results or their dimensional scope.
New
The main manuscript states the sharp constant-curvature comparison in every dimension for finite-volume sets of finite ambient perimeter. It also characterizes bounded positive-volume equality regions in the Euclidean comparison. The companion manuscript states a sharp Euclidean filling bound for integral cycles of dimension at least two in proper CAT(0) spaces, without restricting integer multiplicities or ambient dimension.
Limits and assumptions
The manifold comparison requires a complete, simply connected smooth manifold with sectional curvature bounded above by a constant , and sets with finite volume and finite ambient perimeter. The stated equality characterization is for bounded positive-volume sets in the Euclidean comparison and holds up to null sets. The companion filling statement requires proper CAT(0) spaces and compactly supported integral cycles of dimension ; no claim for lower-dimensional cycles is supplied. The main result has a Lean formalization, but the supplied material does not delineate its exact scope or establish that every equality and companion claim is formalized. The abstracts identify no remaining open questions.
Mini glossary
- Isoperimetric comparison —
- A comparison between the boundary size of a region and that of a reference region with the same volume.
- Sharp bound —
- A bound with the optimal benchmark, here supplied by Euclidean or constant-curvature model geometry.
- Equality rigidity —
- A characterization that forces regions attaining equality to have a specified geometry.
- Null set —
- A set of zero volume, which is disregarded in the stated characterization of equality regions.
- Isometric —
- Having the same metric geometry, as required between an equality region with its induced metric and a round Euclidean ball.
Keywords : Cartan–Hadamard isoperimetry · Equality rigidity · Integral fillings
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most κ ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for κ = 0 are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral n-cycles, n ≥ 2, in arbitrary proper CAT spaces.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity
23/09/2026We resolve the generalized Cartan–Hadamard isoperimetric conjecture in every dimension. In a complete simply connected smooth manifold with sectional curvature at most κ ≤ 0, every finite-volume set of finite ambient perimeter has perimeter at least that of the equal-volume ball in curvature κ. 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.
Sharp integral fillings in CAT(0) spaces
—-spaces-September-23-2026/paper.pdf) Every compactly supported integral n-cycle, n ≥ 2, in a proper CAT(0) space bounds a compactly supported integral current with the sharp Euclidean mass bound. The theorem allows arbitrary integer multiplicities and unrestricted ambient dimension. In particular, we prove the Euclidean Cartan–Hadamard isoperimetric conjecture in dimensions at least three.