Filling inequalities do not depend on topology

TL;DR

This paper demonstrates that in dimensions three and higher, the optimal constants in Gromov's filling inequalities are determined solely by dimension and orientability, not by the specific manifold, contrasting with systolic inequalities.

Contribution

It establishes that filling inequalities' optimal constants are independent of the manifold in higher dimensions, depending only on dimension and orientability.

Findings

Optimal constants depend only on dimension and orientability in dimensions ≥ 3

Contrasts with systolic inequalities that depend on the manifold

Provides new insight into geometric inequalities in Riemannian geometry

Abstract

Gromov's universal filling inequalities relate the filling radius and the filling volume of a Riemannian manifold to its volume. The main result of the present article is that in dimensions at least three the optimal constants in the filling inequalities depend only on dimension and orientability, not on the manifold itself. This contrasts with the analogous situation for the optimal systolic inequality, which does depend on the manifold.