Journal of the
Korean Mathematical Society JKMS

In this paper, we study vanishing properties for $L^2$ harmonic $1$-forms on a gradient shrinking Ricci soliton. We prove that if $(M, g, f)$ is a complete oriented noncompact gradient shrinking Ricci soliton with potential function $f$, then there are no non-trivial $L^2$ harmonic $1$-forms which are orthogonal to $df$. Second, we show that if the scalar curvature of the metric $g$ is greater than or equal to $(n-2)/2$, then there are no non-trivial $L^2$ harmonic $1$-forms on $(M, g)$. We also show that any multiplication of the total differential $df$ by a function cannot be an $L^2$ harmonic $1$-form unless it is trivial. Finally, we derive various integral properties involving the potential function $f$ and $L^2$ harmonic $1$-forms, and handle their applications.

Keywords: gradient shrinking Ricci solitons, $L^2$ harmonic forms

MSC numbers: Primary 53C20; 53C25