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 $${\alpha }^{-1}(\sum _{i=1}^{[\alpha ]}{\lambda }_i+(\alpha -[\alpha ]){\lambda }_{[\alpha ]+1})\ge -\theta \bar{\lambda }$$ for some real number $\alpha \in [1,(n+2)(n-1)\mathrm{/}2)$. Here $[\alpha ]:=\max \{m\in \mathbb{Z}:m\le \alpha \}$, $\theta >-1$ and ${\lambda }_1\le \cdots \le {\lambda }_{(n+2)(n-1)\mathrm{/}2}$ are the eigenvalues of $\mathring{R}$ and $\bar{\lambda }$ 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 $\alpha \in {\mathbb{Z}}_{+}$ of the authors \cite{CW24-1,CW25-2} and Fu-Lu \cite{FL25}.