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September 2017 On the singularities of the Szegő projections on lower energy forms
Chin-Yu Hsiao, George Marinescu
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J. Differential Geom. 107(1): 83-155 (September 2017). DOI: 10.4310/jdg/1505268030

Abstract

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1 , n \geqslant 2$. Let $\Box^{(q)}_{b}$ be the Gaffney extension of Kohn Laplacian on $(0, q)$-forms. We show that the spectral function of $\Box^{(q)}_{b}$ admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if $X$ is compact and the Levi form is non-degenerate of constant signature on $X$, then the spectrum of $\Box^{(q)}_{b}$ in $] 0, \infty [$ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation of the associated Szegő kernel admits an asymptotic expansion under a local closed range condition. As applications, we establish the Szegő kernel asymptotic expansions on some weakly pseudoconvex CR manifolds and on CR manifolds with transversal CR $S^1$ actions. By using these asymptotics, we establish some local embedding theorems on CR manifolds and we give an analytic proof of a theorem of Lempert asserting that a compact strictly pseudoconvex CR manifold of dimension three with a transversal CR $S^1$ action can be CR embedded into $\mathbb{C}^N$, for some $N \in \mathbb{N}$.

Funding Statement

The first author was partially supported by Taiwan Ministry of Science of Technology project 103-2115-M-001-001, the DFG project MA 2469/2-2 and funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative.
The second author partially supported by the DFG projects SFB/TR 12, MA 2469/2-1 and Université Paris 7.

Citation

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Chin-Yu Hsiao. George Marinescu. "On the singularities of the Szegő projections on lower energy forms." J. Differential Geom. 107 (1) 83 - 155, September 2017. https://doi.org/10.4310/jdg/1505268030

Information

Received: 8 March 2015; Published: September 2017
First available in Project Euclid: 13 September 2017

zbMATH: 06846962
MathSciNet: MR3698235
Digital Object Identifier: 10.4310/jdg/1505268030

Rights: Copyright © 2017 Lehigh University

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Vol.107 • No. 1 • September 2017
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