Max R. Jahnke | Searcharxiv

arXiv2026

Real involutive systems on compact Lie groups

On a compact connected Lie group $G$, we study the global solvability and the cohomology spaces of the differential complex associated with an essentially real involutive structure that is invariant under left translations. We prove that solvability in the first degree of the complex implies solvability in all other degrees, and furnish a converse for this fact under a certain commutativity hypothesis (that always holds when $G$ is a torus). Additionally, it is proved that the solvability holds when the structure comes from the Lie algebra of a closed subgroup of $G$. We also investigate real tube structures when $G$ is the base manifold.

math.AP↗

arXiv2023

Global solvability and cohomology of tube structures on compact manifolds

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.

math.AP↗