The CR Almost Schur Lemma and the positivity conditions

TL;DR

This paper proves a new estimate for the pseudohermitian scalar curvature on compact CR manifolds, linking it to the Webster Ricci tensor and torsion, with improved bounds in the torsion-free case.

Contribution

It introduces a novel version of the CR almost Schur Lemma that provides sharper curvature estimates under positivity conditions.

Findings

Establishes a new curvature estimate for pseudohermitian manifolds.

Shows the estimate improves in the torsion-free (Sasakian) case.

Connects positivity conditions with scalar curvature constancy.

Abstract

We establish a new version of the CR almost Schur Lemma which gives an estimation of the pseudohermitian scalar curvature on a compact strictly pseudoconvex pseudohermitian manifold to be a constant in terms of the norm of the traceless Webster Ricci tensor and the pseudohermitian torsion under a certain positivity condition. In the torsion-free case, i.e. for a compact Sasakian manifold, our positivity condition coincides with the known one and we obtain a better estimate