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

In this note, we study Einstein manifolds whose curvature operator of the second kind $\mathring{R}$ satisfies the cone condition \[ α^{-1}\big(\sum_{i=1}^{[α]} λ_i+ (α- [α] ) λ_{[α] + 1} \big) \ge -θ\barλ \] for some real number $α\in [1, (n+2)(n-1)/2)$. Here $[α] :=\max\{ m \in \mathbb{Z}: m \leq α\}$, $θ>-1$ and $λ_1 \le \cdots \le λ_{(n+2)(n-1)/2}$ 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}.