Abstract
In this paper, we first prove the CR analogue of Obata’s theorem on a closed pseudohermitian 3-manifold with zero pseudohermitian torsion. Secondly, instead of zero torsion, we have the CR analogue of Li-Yau’s eigenvalue estimate on the lower bound estimate of positive first eigenvalue of the sub-Laplacian in a closed pseudohermitian 3-manifold with nonnegative CR Paneitz operator P 0. Finally, we have a criterion for the positivity of first eigenvalue of the sub-Laplacian on a complete noncompact pseudohermitian 3-manifold with nonnegative CR Paneitz operator. The key step is a discovery of integral CR analogue of Bochner formula which involving the CR Paneitz operator.
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This research was supported in part by the NSC of Taiwan.
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Chang, SC., Chiu, HL. On the CR analogue of Obata’s theorem in a pseudohermitian 3-manifold. Math. Ann. 345, 33–51 (2009). https://doi.org/10.1007/s00208-009-0339-3
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DOI: https://doi.org/10.1007/s00208-009-0339-3