Rigidity of eigenvalues of shrinking Ricci solitons

Abstract

In this paper, we study the rigidity of eigenvalues of shring Ricci solitons. It is known that the drifted Laplacian on shrinking Ricci solitons has discrete spectrum, its eigenvalues have a lower bound and a rigidity result holds. Firstly, we show that if the $n^{\text{th}}$ eigenvalue is close to this lower bound, then the $n$-soliton must be the trivial Gaussian soliton ${\mathbb{R}}^n$. Secondly, we show similar results for the $(n-1{)}^{\text{th}}$ and $(n-2{)}^{\text{th}}$ eigenvalue under a non-collapsing condition. Lastly, we give an alomost rigidity for the $k^{\text{th}}$ eigenvalue with general $k$. Part of our results could be viewed as an soliton (could be noncompact) analog of Theorem 1.1 (which only holds for compact manifolds) in Peterson (Invent. Math. 138 (1999): 1-21).