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).