Einstein manifolds under cone conditions for the curvature operator of the second kind

It is established in [6, 14, 23] that any closed Einstein manifold with two-nonnegative curvature operator of the second kind is either flat or a round sphere. In this paper, we refine this result by relaxing the curvature condition to a cone condition (strictly weaker than two nonnegativity) proposed by Li [18]. Precisely, we prove that any closed Einstein manifold of dimension $n=4$ or $n=5$ or $n\ge 8$, if the curvature operator of the second kind $\mathring{R}$ satisfies \begin{align*} (λ_1+λ_2)/2 \ge -θ(n) \bar λ, \end{align*} then the manifold is either flat or a round sphere. Here, ${\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_{(n-1)(n+2)\mathrm{/}2}$ are the eigenvalues of $\mathring{R}$, $\bar{\lambda }$ is their average, and $\theta (n)$ is a positive constant defined as in (1.2).