Stability of isoperimetric and Sobolev inequalities for excluded exponents on Riemannian manifolds

Abstract Quasi-isometries are a versatile type of maps that preserve the large-scale geometry of spaces, while introducing significant local distortions. Following Kanai’s work, which established the invariance of various analytic and geometric properties under quasi-isometries, this paper generalizes isoperimetric and Sobolev inequalities for exponents less than the manifold’s dimension, proving both that they are equivalent and preserved by quasi-isometries.