Gromov’s isoperimetric gap conjecture for Hadamard spaces

Establish Gromov’s isoperimetric gap conjecture for general Hadamard spaces by proving that the optimal isoperimetric filling exponent changes from the Euclidean exponent 1+1/k to the linear exponent 1 precisely when the cycle dimension k reaches the asymptotic rank of the space.

Background

The paper studies filling inequalities for integral cycles in complete CAT(0)-spaces, also called Hadamard spaces. In Euclidean-type behavior, a k-cycle admits a filling whose mass is controlled by a power of its mass with exponent 1+1/k, whereas hyperbolic or higher-rank behavior can yield a linear filling estimate. The asymptotic rank is used to identify the dimension at which this transition should occur.

The paper proves the conjectured linear inequality for Hadamard spaces with finite asymptotic Nagata dimension and finite asymptotic rank, for all cycle dimensions at least the asymptotic rank. The broader conjecture, without these additional dimensional and covering assumptions, is not resolved by the stated result and is therefore included as an explicitly identified unresolved conjecture.