Einstein manifolds of negative lower bounds on curvature operator of the second Kind

We demonstrate that $n$-dimension closed Einstein manifolds, whose smallest eigenvalue of the curvature operator of the second kind of $\mathring{R}$ satisfies $\lambda_1 \ge -\theta(n) \bar\lambda$, are either flat or round spheres, where $\bar \lambda$ is the average of the eigenvalues of $\mathring{R}$, and $\theta(n)$ is defined as in equation (1.2). Our result improves a celebrated result (Theorem 1.1) concerning Einstein manifolds with nonnegative curvature operator of the second kind.
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