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We revisit some uniqueness results for a geometric nonlinear PDE related tothe scalar curvature in Riemannian geometry and CR geometry. In the Riemanniancase we give a new proof of the uniqueness result assuming only a positivelower bound for Ricci curvature. We apply the same principle in the CR case andreconstruct the Jerison-Lee identity in a more general setting. As aconsequence we prove a stronger uniqueness result in the CR case. We alsodiscuss some open problems for further study.

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