A mathematical preprint reports a result addressing Gromov’s predicted linear isoperimetric filling inequality for Hadamard spaces at or above their asymptotic rank. It states that complete CAT(0) spaces with finite asymptotic Nagata dimension and finite asymptotic rank satisfy a linear filling inequality in cycle dimensions k >= max{nu, 1}. In plain language, certain higher-dimensional boundaries can be filled by objects whose mass grows no faster than a constant times the mass of the boundary.
The work is theoretical and deals with finite-mass integral currents, with compactly supported integral currents as a subgroup. The central relationship is between the mass of a cycle and the mass of its filling. The decomposition constants depend on k and quantitative Nagata-cover data, rather than on an individual current, simplex number or scale.
Where the threshold begins
The threshold is the key qualification. The main statement concerns an integral k-cycle with finite mass; when k is at least the asymptotic-rank threshold, the filling mass is bounded linearly by the cycle’s mass. The constant exists for the stated space and dimension, but the result does not supply one numerical value for every possible space.
The same proof strategy first establishes a self-improvement result. If a filling-volume estimate has the specified sub-Euclidean growth, the theorem converts it into a linear filling bound for integral cycles. It also preserves compact support: a compactly supported cycle can receive a filling with compact support.
The author presents this as an upgrade of cited rank-one and rank-two results whose exponents could be made arbitrarily close to one, to the optimal linear exponent. The stated approach also covers arbitrary finite asymptotic ranks, rather than stopping at those two cases.
How the proof works
The proof’s key move is a one-scale deformation. It discretizes a minimizing filling and deforms it at one chosen scale, rather than relying on a repeated procedure across many scales. At the level of the argument, the resulting simplicial count gains one inverse power of the scale, a change that supports the linear estimate in filling dimension k+1.
A central decomposition writes a compactly supported cycle as T = Q + ∂S. The part Q has its own compactly supported filling, and the associated filling-function term becomes absorbable once the scale is sufficiently large.
The absorption step chooses a scale s* for which the relevant coefficient alpha(s*) is at most 1/2. The resulting linear estimate is displayed with coefficient 2a0s*, and the argument then promotes the compact-support result to finite-mass cycles.
That final promotion uses Wenger’s thick-thin decomposition. The paper says this part does not require finite asymptotic Nagata dimension, even though that dimension condition is required elsewhere in the main theorem.
Consequences within the framework
The paper draws a further structural conclusion within the same class of spaces: the asymptotic rank nu is the first cycle dimension k >= 1 whose filling function has linear O(r) growth. That gives the rank a characterization in terms of when linear filling begins.
One application is stated for oriented Hadamard manifolds of dimension n >= 2 whose asymptotic rank is at most n - 1 and whose asymptotic Nagata dimension is finite. In that setting, every bounded finite-perimeter set satisfies a volume-perimeter inequality, vol(E) <= C Per(E), with corresponding Cheeger and spectral lower-bound consequences.
Another application gives a relative deformation estimate under additional compactness, quasiconvexity and (LCk) or (EIk) assumptions. The compactly supported relative deformation obeys the stated boundary identities, and its total mass is bounded linearly by the mass of the original current.
What remains outside the claim
The result remains conditional. The main theorem requires complete CAT(0) structure, finite asymptotic Nagata dimension and finite asymptotic rank, and its principal linear inequality is stated only for cycle dimensions at or above the asymptotic-rank threshold. It does not extend the conclusion to spaces without those assumptions.
The paper identifies the finite simplicial approximation step as the main bottleneck for possible improvements to the Nagata-dimension condition.
The document is a preprint identified as arXiv:2608.26059v1 [math.MG], dated 26 Aug 2026. The author acknowledges partial support from the German Research Foundation through grants FR 2664/3-1 and TRR 352-Project-ID 470903074, as well as from the Studienstiftung des deutschen Volkes.
Paper data and sources
Original title: Linear isoperimetric filling inequalities in Hadamard spaces at and above the asymptotic rank
Authors: Jonas W. Peteranderl
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text