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**A new characterization of the CR sphere and the sharp eigenvalue estimate for the Kohn Laplacian.**
*(English)*
Zbl 1319.53032

Summary: In Riemannian geometry, the well-known Lichnerowicz-Obata theorem gives a sharp estimate and a characterization of the equality case (rigidity theorem) for the first positive eigenvalue of the Laplacian. In CR geometry, analogous problems are more delicate due to the presence of the torsion of Tanaka-Webster connection. Estimates and rigidity theorems for the first positive eigenvalue of the sub-Laplacian have been studied by many authors (see [J. Differ. Geom. 95, No. 3, 483–502 (2013; Zbl 1277.32038)] and the reference therein), e.g., S.-Y. Li and X. Wang proved an Obata-type theorem for sub-Laplacian without any additional assumption on the torsion.

In [Duke Math. J. 161, No. 15, 2909–2921 (2012; Zbl 1271.32040)] S. Chanillo et al. gave a sharp eigenvalue estimate for the Kohn Laplacian on three-dimensional manifolds. This poses the question whether a version of rigidity theorem holds in this case. A partial answer was given by S.-C. Chang and C.-T. Wu [“On the CR Obata theorem for Kohn Laplacian in a closed pseudohermitian hypersurface in \(\mathbb C^{n+1}\)”, Preprint] in 2012. They generalized the eigenvalue estimate to general dimension and proved, under additional assumptions on the torsion, that the equality holds only when the manifold is the CR sphere. In the present paper, we completely resolve the rigidity question for manifolds of dimension at least five by establishing a new characterization of the CR sphere in terms of the existence of a (non-trivial) function satisfying a certain overdetermined system. The result holds without any assumption on the pseudo-Hermitian torsion.

In [Duke Math. J. 161, No. 15, 2909–2921 (2012; Zbl 1271.32040)] S. Chanillo et al. gave a sharp eigenvalue estimate for the Kohn Laplacian on three-dimensional manifolds. This poses the question whether a version of rigidity theorem holds in this case. A partial answer was given by S.-C. Chang and C.-T. Wu [“On the CR Obata theorem for Kohn Laplacian in a closed pseudohermitian hypersurface in \(\mathbb C^{n+1}\)”, Preprint] in 2012. They generalized the eigenvalue estimate to general dimension and proved, under additional assumptions on the torsion, that the equality holds only when the manifold is the CR sphere. In the present paper, we completely resolve the rigidity question for manifolds of dimension at least five by establishing a new characterization of the CR sphere in terms of the existence of a (non-trivial) function satisfying a certain overdetermined system. The result holds without any assumption on the pseudo-Hermitian torsion.

### MSC:

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

32V30 | Embeddings of CR manifolds |

### References:

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[10] | Kohn, J. J., Boundaries of complex manifolds, (Proc. Conf. Complex Manifolds, Minneapolis, 1964, (1965), Springer-Verlag New York), 81-94 |

[11] | Li, S.-Y.; Wang, X., An obata-type theorem in CR geometry, J. Differential Geom., 95, 3, 483-502, (2013) · Zbl 1277.32038 |

[12] | Obata, M., Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan, 14, 333-340, (1962) · Zbl 0115.39302 |

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