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 \((M,\Theta)\).


32V20 Analysis on CR manifolds
32W10 \(\overline\partial_b\) and \(\overline\partial_b\)-Neumann operators
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