Global solvability and cohomology of tube structures on compact manifolds

TL;DR

This paper develops new techniques using partial Fourier series to analyze the solvability and cohomology of differential complexes on manifolds with tube structures, generalizing previous results and solving open problems.

Contribution

It introduces a systematic approach to characterize global solvability and cohomology of tube structures, extending existing theories to complex and real cases on compact manifolds.

Findings

Complete characterization of global solvability for complex tube structures.

New results on the finiteness of cohomology spaces in intermediate degrees.

Necessary and sufficient conditions for the differential operator to have closed range.

Abstract

We introduce new techniques to study the differential complexes associated to tube structures on $M \times \mathbb{T}^m$ of corank $m$, in which $M$ is a compact manifold and $\mathbb{T}^m$ is the $m$-torus. By systematically employing partial Fourier series, for complex tube structures, we completely characterize global solvability, in a given degree, in terms of a weak form of hypoellipticity, thus generalizing existing results and providing a broad answer to an open problem proposed by Hounie and Zugliani (2017). We also obtain new results on the finiteness of the cohomology spaces in intermediate degrees. In the case of real tube structures, we extend an isomorphism for the cohomology spaces originally obtained by Dattori da Silva and Meziani (2016) in the case $M = \mathbb{T}^n$. Moreover, we establish necessary and sufficient conditions for the differential operator to have closed range in the first degree.