## The Webster scalar curvature and sharp upper and lower bounds for the first positive eigenvalue of the Kohn-Laplacian on real hypersurfaces.(English)Zbl 1404.32071

The authors study upper and lower bounds for the first positive eigenvalue $$\lambda_1$$ of the Kohn-Laplacian on compact strictly presudoconvex hypersurfaces $$(M,\Theta)$$ in $$\mathbb C^{n+1}$$.
In particular, they provide a sharp upper bound for $$\lambda_1$$ given explicitly in terms of the defining function $$\rho$$ under a technical assumption on $$\rho$$. As a corollary, a Reilly-type estimate follows when $$M$$ is embedded into the unit sphere.
Also, when the structure $$\Theta$$ on $$M\subset\mathbb C^{n+1}$$ is (the unique) volume-normalized with respect to $$dz^1\wedge dz^2\wedge\cdots\wedge dz^{n+1}$$, they establish a lower bound for $$\lambda_1$$ given explicitly in terms of the Webster scalar curvature of $$(M,\Theta)$$.

### MSC:

 32V20 Analysis on CR manifolds 32W10 $$\overline\partial_b$$ and $$\overline\partial_b$$-Neumann operators
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### References:

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