Uniqueness Results on a geometric PDE in Riemannian and CR Geoemetry Revisited

TL;DR

This paper revisits and strengthens uniqueness results for a geometric nonlinear PDE related to scalar curvature in Riemannian and CR geometries, providing new proofs and generalizations.

Contribution

It offers a new proof of uniqueness in Riemannian geometry under Ricci curvature bounds and extends the Jerison-Lee identity to more general CR settings.

Findings

New proof of uniqueness assuming Ricci curvature lower bound

Generalized Jerison-Lee identity in CR geometry

Stronger uniqueness results in CR geometry

Abstract

We revisit some uniqueness results for a geometric nonlinear PDE related to the scalar curvature in Riemannian geometry and CR geometry. In the Riemannian case we give a new proof of the uniqueness result assuming only a positive lower bound for Ricci curvature. We apply the same principle in the CR case and reconstruct the Jerison-Lee identity in a more general setting. As a consequence we prove a stronger uniqueness result in the CR case. We also discuss some open problems for further study.