Cone Conditions for the Curvature Operator of the Second Kind on Einstein Manifolds

Abstract: In this note, we study Einstein manifolds whose curvature operator of the second kind $\mathring{R}$ satisfies the cone condition [ α^{-1}\big(\sum_{i=1}^{[α]} λi+ (α- [α] ) λ{[α] + 1} \big) \ge -θ\barλ ] for some real number $α\in [1, (n+2)(n-1)/2)$. Here $[α] :=\max{ m \in \mathbb{Z}: m \leq α}$, $θ&gt;-1$ and $λ<em>1 \le \cdots \le λ</em>{(n+2)(n-1)/2}$ are the eigenvalues of $\mathring{R}$ and $\barλ$ is their average. The main result states that any closed Einstein manifold of dimension $n \ge 4$ with $\mathring{R}$ satisfies the cone condition is flat or a round sphere. These results generalize recent works corresponding to $α\in \mathbb Z_+$ of the authors \cite{CW24-1,CW25-2} and Fu-Lu \cite{FL25}.