Abstract. We investigate tensor products of Hilbert complexes, in particular the (essential) spectrum of their Laplacians. It is shown that the essential spectrum of the Laplacian associated to the tensor product complex is computable in terms of the spectra of the factors. Applications are given for the ∂-Neumann problem on the product of two or more Hermitian manifolds, especially regarding (non-) compactness of the associated ∂-Neumann operator.
Given a smooth positive measure µ on a complete Hermitian manifold with Ricci curvature bounded from below, we prove a pointwise Agmon-type bound for the corresponding Bergman kernel, under rather general conditions involving the coercivity of an associated complex Laplacian on (0, 1)-forms. Thanks to an appropriate version of the Bochner-Kodaira-Nakano basic identity, we can give explicit geometric sufficient conditions for such coercivity to hold.Our results extend several known bounds in the literature to the case in which the manifold is neither assumed to be Kähler nor of "bounded geometry". The key ingredients of our proof are a localization formula for the complex Laplacian (of the kind used in the theory of Schrödinger operators) and a mean value inequality for subsolutions of the heat equation on Riemannian manifolds due to Li, Schoen, and Tam.We also show in an appendix that the "twisted basic identities" of, e.g., [MV15] are standard basic identities with respect to conformally Kähler metrics.
We study necessary conditions for compactness of the weighted ∂-Neumann operator on the space L 2 (C n , e −ϕ ) for a plurisubharmonic function ϕ. Under the assumption that the corresponding weighted Bergman space of entire functions has infinite dimension, a weaker result is obtained by simpler methods. Moreover, we investigate (non-) compactness of the ∂-Neumann operator for decoupled weights, which are of the form ϕ(z) = ϕ1(z1) + · · · + ϕn(zn). More can be said if every ∆ϕj defines a nontrivial doubling measure.2010 Mathematics Subject Classification. Primary 32W05; Secondary 30H20, 35N15, 35P10.
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