Sharp Caffarelli-Kohn-Nirenberg inequalities on Riemannian manifolds: the influence of curvature

TL;DR

This paper establishes sharp Caffarelli-Kohn-Nirenberg inequalities on Euclidean and Riemannian manifolds, demonstrating how curvature influences these inequalities and providing rigidity results for manifolds supporting them.

Contribution

It extends sharp CKN inequalities to Cartan-Hadamard manifolds and introduces curvature-dependent remainder terms, broadening Kristály's recent results.

Findings

Sharp CKN inequalities on Euclidean spaces

Extension to Cartan-Hadamard manifolds with same best constants

Rigidity results linking curvature and inequality support

Abstract

We first establish a family of sharp Caffarelli-Kohn-Nirenberg type inequalities on the Euclidean spaces and then extend them to the setting of Cartan-Hadamard manifolds with the same best constant. The quantitative version of these inequalities also are proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli-Kohn-Nirenberg type inequalities with the same sharp constant as in $\mathbb R^n$ (shortly, sharp CKN inequalities). Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Krist\'aly to a larger class of the sharp CKN inequalities.