Global hypoellipticity for involutive systems on non-compact manifolds

TL;DR

The paper addresses global hypoellipticity for involutive first-order systems on non-compact manifolds of the form $M\times\mathbb{T}^m$, where $M$ has a scattering (asymptotically Euclidean) metric. It develops an arithmetic-geometry framework based on Hodge theory for scattering manifolds and partial Fourier analysis to reduce the problem to modewise (and then linear-algebraic) criteria. For $m=1$ the operator is globally hypoelliptic precisely when the closed 1-form $\omega$ is neither rational nor Liouville; for $m>1$ the criterion generalizes via the cycle matrix $A(\boldsymbol{\omega})$ and a Diophantine condition, yielding a complete characterization. These results extend compact-case findings to non-compact scattering geometries and illustrate a deep link between arithmetic properties of cohomology data and global regularity. The work has potential implications for regularity theory on asymptotically flat spaces and related geometric-analytic problems on non-compact manifolds.

Abstract

We study the global hypoellipticity of the operator $\mathbb{L} = \mathrm{d}_t + \sum_{k=1}^m ω_k \wedge \partial_{x_k}$, 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 $ω_1,\dots,ω_m$ 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 $ω_k$. The analysis relies on microlocal techniques adapted to the scattering setting and a version of the Hodge Theorem for scattering manifolds.

Theorems & Definitions (30)

• Definition 1.1
• Proposition 2.1
• proof
• Remark
• Theorem 2.2: Hodge Theorem for Asymptotically Euclidean Manifolds
• Remark
• Theorem 2.3
• Proposition 2.4
• proof
• Corollary 2.5