Quantitative stability of Sobolev inequalities on compact Riemannian manifolds

TL;DR

This paper establishes quantitative stability results for Sobolev inequalities on compact Riemannian manifolds, showing near-extremal functions are close to extremals in Sobolev space, with implications for sub-critical cases and degenerate phenomena.

Contribution

It provides new quantitative stability estimates for Sobolev inequalities on manifolds, extending previous results to a broader class and including sub-critical cases.

Findings

Near-saturation implies closeness to extremal functions in Sobolev space

Stability results depend on manifold-dependent constants

Degenerate phenomena are identified and discussed

Abstract

We study quantitative stability results for different classes of Sobolev inequalities on general compact Riemannian manifolds. We prove that, up to constants depending on the manifold, a function that nearly saturates a critical Sobolev inequality is quantitatively $W^{1,2}$-close to a non-empty set of extremal functions, provided that the corresponding optimal Sobolev constant satisfies a suitable strict bound. The case of sub-critical Sobolev inequalities is also covered. Finally, we discuss degenerate phenomena in our quantitative controls.