## 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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