$L^{2}$ harmonic forms on complete special holonomy manifolds

TL;DR

This paper proves that on complete special holonomy manifolds with certain structures, all square-integrable harmonic 2-forms vanish, leading to trivial solutions for specific gauge equations, advancing understanding of geometric analysis in these contexts.

Contribution

It establishes vanishing results for $L^{2}$ harmonic 2-forms on complete $G_{2}$ and $Spin(7)$ manifolds with linear structures, and applies this to gauge theory.

Findings

$L^{2}$ harmonic 2-forms vanish on the specified manifolds.

Trivial solutions for the instanton equation with square integrable curvature.

Extension of Hodge theory to principal $G$-bundles over these manifolds.

Abstract

In this article, we consider $L^{2}$ harmonic forms on a complete non-compact Riemannian manifold $X$ with a nonzero parallel form $\omega$. The main result is that if $(X,\omega)$ is a complete $G_{2}$- ( or $Spin(7)$-) manifold with a $d$(linear) $G_{2}$- (or $Spin(7)$-) structure form $\omega$, the $L^{2}$ harmonic $2$-forms on $X$ will be vanish. As an application, we prove that the instanton equation with square integrable curvature on $(X,\omega)$ only has trivial solution. We would also consider the Hodge theory on the principal $G$-bundle $E$ over $(X,\omega)$.