Onofri inequalities and rigidity results

TL;DR

This paper advances the understanding of the Moser-Trudinger-Onofri inequality on Riemannian manifolds by establishing rigidity results, estimating optimal constants, and introducing a nonlinear flow approach, with applications to both compact and non-compact cases.

Contribution

It introduces a non-local criterion and a nonlinear flow method to analyze the inequality, providing new estimates and optimality discussions, especially for weighted cases in Euclidean space.

Findings

Established a rigidity result for the Euler-Lagrange equation.

Derived an estimate of the optimal constant in the inequality.

Extended results to weighted inequalities in Euclidean space, relevant for chemotaxis models.

Abstract

This paper is devoted to the Moser-Trudinger-Onofri inequality on smooth compact connected Riemannian manifolds. We establish a rigidity result for the Euler-Lagrange equation and deduce an estimate of the optimal constant in the inequality on two-dimensional closed Riemannian manifolds. Compared to existing results, we provide a non-local criterion which is well adapted to variational methods, introduce a nonlinear flow along which the evolution of a functional related with the inequality is monotone and get an integral remainder term which allows us to discuss optimality issues. As an important application of our method, we also consider the non-compact case of the Moser-Trudinger-Onofri inequality on the two-dimensional Euclidean space, with weights. The standard weight is the one that is computed when projecting the two-dimensional sphere using the stereographic projection, but we also give more general results which are of interest, for instance, for the Keller-Segel model in chemotaxis.