L2 HARMONIC FORMS ON GRADIENT SHRINKING RICCI SOLITONS
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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.
Abstract
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.
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