Essential spectra of tensor product Hilbert complexes, and the $\overline\partial$-Neumann problem on product manifolds

TL;DR

This paper studies the spectra of tensor product Hilbert complexes, especially the essential spectrum of Laplacians, and applies these results to the $ar{ ext{d}}$-Neumann problem on product Hermitian manifolds, focusing on operator compactness.

Contribution

It provides a method to compute the essential spectrum of tensor product Laplacians using the spectra of individual factors, advancing understanding of the $ar{ ext{d}}$-Neumann problem on product manifolds.

Findings

Essential spectrum of tensor product Laplacians is explicitly computable from factors.

Criteria for compactness of the $ar{ ext{d}}$-Neumann operator on product manifolds.

Applications to non-compactness scenarios in complex analysis.

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 $\overline\partial$-Neumann problem on the product of two or more Hermitian manifolds, especially regarding (non-) compactness of the associated $\overline\partial$-Neumann operator.