Extremals for Logarithmic Hardy-Littlewood-Sobolev inequalities on compact manifolds

TL;DR

This paper investigates extremal metrics on compact manifolds that minimize a regularized trace of the inverse Laplace-Beltrami operator, establishing bounds and existence results related to logarithmic Hardy-Littlewood-Sobolev inequalities and their duals.

Contribution

It extends extremal metric results and Hardy-Littlewood-Sobolev inequalities from surfaces to general compact manifolds, including existence and optimality conditions.

Findings

The infimum of the regularized trace is less than or equal to that of the round sphere.

The infimum is attained if it equals the sphere's value.

Results generalize to higher-dimensional compact manifolds.

Abstract

For a closed connected surface with a metric g, we consider the regularized trace of the inverse of the Laplace-Beltrami operator. We minimize this on the class of smooth metrics conformal to g having the same area, and show that the infimum is less than or equal to the value for the round sphere of the same area, and if it is equal, then it is attained. In fact we prove the analogs of these results for general dimensional compact manifolds. Explicitly, the results are logarithmic Hardy-Littlewood-Sobolev inequalites. By duality they give analogs of the Onofri-Beckner theorem.