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 corollary, we prove a more general uniqueness result for a family of semilinear PDE on a closed pseudohermitian manfiold with zero torsion and positive Ricci curvature. We also discuss some open problems for further study.
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Acknowledgements
The author is very grateful to the referee for an extremely careful reading of the manuscript and for making many suggestions which have greatly improved the presentation. The author is partially supported by Simons Foundation Collaboration Grant for Mathematicians #312820.
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Wang, X. Uniqueness results on a geometric PDE in Riemannian and CR geometry revisited. Math. Z. 301, 1299–1314 (2022). https://doi.org/10.1007/s00209-021-02963-9
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DOI: https://doi.org/10.1007/s00209-021-02963-9