Global hypoellipticity for involutive systems on non-compact manifolds

TL;DR

This paper investigates the conditions under which a differential operator on non-compact manifolds with scattering metrics is globally hypoelliptic, extending known results from compact to non-compact settings using microlocal analysis.

Contribution

It extends the characterization of global hypoellipticity for involutive systems to non-compact manifolds with scattering metrics, incorporating arithmetic conditions and microlocal techniques.

Findings

Characterization of global hypoellipticity in non-compact scattering manifolds.

Extension of previous compact manifold results to non-compact settings.

Use of microlocal analysis and scattering Hodge theory in the proofs.

Abstract

We study the global hypoellipticity of the operator $\mathbb{L} = \mathrm{d}_t + \sum_{k=1}^m \omega_k \wedge \partial_{x_k}$, defined on differential forms over product manifolds of the form $M \times \mathbb{T}^m$, where $M$ is a non-compact manifold homeomorphic to the interior of a compact manifold with boundary, equipped with a scattering metric, and $\omega_1,\dots,\omega_m$ are smooth closed 1-forms on $M$. Extending previous results obtained in the compact setting, we characterize global hypoellipticity of $\mathbb{L}$ in terms of arithmetic properties of the forms $\omega_k$. The analysis relies on microlocal techniques adapted to the scattering setting and a version of the Hodge Theorem for scattering manifolds.