$L^2$ harmonic forms on complete special holonomy 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 $(X,\omega )$.

To appear in AGAG

References in corpus (3)

• Gauge Theory in higher dimensions, II
• Instantons on Special Holonomy Manifolds
• Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps