Rigidity of eigenvalues of shrinking Ricci solitons

TL;DR

This paper investigates the rigidity properties of eigenvalues of the drifted Laplacian on shrinking Ricci solitons, establishing conditions under which the soliton must be trivial or nearly so, extending known results to noncompact cases.

Contribution

It provides new rigidity results for eigenvalues of the drifted Laplacian on shrinking Ricci solitons, including almost rigidity for general eigenvalues and noncompact analogs of classical theorems.

Findings

Eigenvalues near the lower bound imply the soliton is Gaussian $\

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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).