Riemannian manifolds with positive Yamabe invariant and Paneitz operator

TL;DR

This paper establishes a fundamental equivalence between the existence of a conformal metric with positive scalar and Q curvature and the positivity of the Yamabe invariant and Paneitz operator on high-dimensional Riemannian manifolds.

Contribution

It proves that for manifolds of dimension six or higher, positive scalar and Q curvature conformal metrics exist if and only if both the Yamabe invariant and Paneitz operator are positive, linking geometric and analytical conditions.

Findings

Equivalence between conformal metric existence and positivity conditions

Positive Yamabe invariant and Paneitz operator imply positive scalar and Q curvature

Applicable to manifolds of dimension six and higher

Abstract

For a Riemannian manifold with dimension at least six, we prove that the existence of a conformal metric with positive scalar and Q curvature is equivalent to the positivity of both the Yamabe invariant and the Paneitz operator.