Let X be a Hadamard space and let r=operatornameasrk(X) be its asymptotic rank. For every integer k≥ r, there exists a constant C=C(X,k)>0 such that every integral k-cycle T∈mathbfI_k(X) admits an integral (k+1)-current V∈mathbfI_k+1(X) with partial V=T and mathbfM(V)≤ C,mathbfM(T), where mathbfM denotes mass.

Gromov’s linear isoperimetric filling conjecture for Hadamard spaces

Gromov’s conjecture predicts linear filling bounds for integral cycles in dimensions at least the asymptotic rank of a Hadamard space. The latest claim concerns spaces with finite asymptotic Nagata dimension and finite asymptotic rank.

Known results

Asymptotic rank at most $2$ : near-linear bounds $\mathbf{M}(V)\leq C\mathbf{M}(T)^{1+\delta}$ , not linear, under finite asymptotic Nagata dimension (Lang, Stadler, and Urech, 2025).
Homogeneous Hadamard manifolds: linear bounds for $r\leq k<n$ (2022 preprint).

August 2026 claimed proof

A preprint reported on August 26, 2026 claims linear filling bounds for integral cycles in every dimension $k$ at least the asymptotic rank, under finite asymptotic Nagata dimension and rank. If correct, this extends the rank-one and rank-two theory to arbitrary finite asymptotic ranks; the claim is unrefereed and unverified.

Current status (as of August 2026): the conjecture is claimed solved under finite asymptotic Nagata-dimension and rank assumptions, but the new theorem remains unverified.

Gromov’s linear isoperimetric filling conjecture for Hadamard spaces

Primary source

Additional references

Progress summary

Solutions 0