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Hearing the type of a domain in \(\mathbb C^2\) with the \(\bar{\partial}\)-Neumann Laplacian. (English) Zbl 1156.32023

The author studies the interplay between the geometry of a bounded domain in \(\mathbb{C}^{n}\) and the spectrum of the \(\overline{\partial}\)-Neumann Laplacian. The main result obtained by the author is the following: Let \(\Omega\) be a smooth bounded pseudoconvex domain in \(\mathbb{C}^{2}\); denote by \(N(\lambda)\) the number of eigenvalues of the \(\overline{\partial}\)-Neumann Laplacian less or equal to \(\lambda\). Then \(b\Omega\) is of finite type if \(N(\lambda)\) has at most polynomial growth.
To show that \(N(\lambda)\) has polynomial growth, the author studies the spectral kernel of the \(\overline{\partial}\)-Neumann Laplacian following the analysis of G. Métivier [Duke Math. J. 48, 779–806 (1981; Zbl 0489.35064)]. The spectral kernel (unlike the Bergman kernel) does not behave well under biholomorphic mappings. The author overcomes this difficulty by rescaling both the domain and the \(\overline{\partial}\)-Neumann Laplacian; for this he is led to use anisotropic bidiscs that have larger radii in the complex normal direction. An essential role is played by the result of J. E. Fornaess and N. Sibony [Duke Math. J. 58, No. 3, 633–655 (1989; Zbl 0679.32017)] concerning bidiscs, and Kohn-type uniform subelliptic estimates for the rescaled \(\overline{\partial}\)-Neumann Laplacian. After the flattening of the boundary the problem is reduced to estimate eigenvalues of auxiliary operators in the half-space, and then a semi-classical analysis of Schrödinger operators with finitely degenerated magnetic field [S. Fu and E. Straube, J. Math. Anal. Appl. 271, 267–282 (2002; Zbl 1098.32021) and J. D’Angelo and J. J. Kohn, in: Schneider, Michael (ed.) et al., Several complex variables. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 37, 199–232 (1999; Zbl 0969.32016)].
For the necessity part, the author uses a wavelet construction of P. G. Lemarié and Y. Meyer [Rev. Mat. Iberoam. 2, No. 1-2, 1–18 (1987; Zbl 0657.42028)]. In fact, the author proves a more general result for domains, \(\Omega\) in \(\mathbb{C}^{n}\); in fact this result gives an evaluation of the type of \(b\Omega\). The result proved for any \(n\) uses a result of J. Yu [Trans. Am. Math. Soc. 347, 587–614 (1995; Zbl 0814.32006)].

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32N15 Automorphic functions in symmetric domains
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References:

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