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*} (\lambda_1+\lambda_2)/2 \ge -\theta(n) \bar \lambda, \end{align*} then the manifold is either flat or a round sphere. Here, $\lambda_1\le \lambda_2\le \cdots\le \lambda_{(n-1)(n+2)/2}$ are the eigenvalues of $\mathring{R}$, $\bar \lambda$ is their average, and $\theta(n)$ is a positive constant defined as in (1.2).