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The sharp upper bounds for the first positive eigenvalue of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces. (English) Zbl 1396.32017

Let \(\rho\) be a smooth strictly plurisubharmonic function on \(\mathbb C^{n+1}\) and \(\nu\) a regular value of \(\rho\) such that \(M:=\rho^{-1} (\nu )\) is compact. \(\rho\) induces a pseudohermitian structure \(\theta = (i/2)(\overline \partial \rho - \partial \rho)\), which gives rise to a volume form \(dv:= \theta \wedge (d\theta )^n\) on \(M\). Furthermore, \(\rho\) induces a Kähler metric \(\rho_{j \overline k}dz^j\, d\overline z^k\) in a neighborhood \(U\) of \(M\). Let \((\rho^{j \overline k})^t\) be the inverse of \(\rho_{j \overline k}\). For a smooth function \(u\) on \(U\) the length of \(\partial u\) in the Kähler metric is given by \(|\partial u|^2_\rho = \rho^{j \overline k} u_j \overline u_{\overline k}\). The authors use an expression of the Kohn-Laplacian \(\square_b = \overline \partial_b^* \, \overline \partial_b\) acting on functions in terms of \(\rho\) in order to estimate the first positive eigenvalue \(\lambda_1\) of \(\square_b\) on \(M\). They suppose that there exists \(j\) such that \(\rho_{j \overline k \ell}=0\) for all \(k\) and \(\ell\) and show that
\[ \lambda_1 \leq \frac{n}{v(M)} \int_M |\partial \rho |^{-2}_\rho \, \theta \wedge (d\theta)^n, \] where \(v(M)= \int_M \theta \wedge (d\theta)^n\) denotes the volume of \(M\). They also prove that equality in the estimate of \(\lambda_1\) occurs only if \(|\partial \rho |^{2}_\rho \) is constant on \(M\), which implies that \(M\) must be a sphere. In addition, they show that on real ellipsoids, the upper bound for \(\lambda_1\) can be computed explicitly.

MSC:

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