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

Abstract

We demonstrate that $n$-dimension closed Einstein manifolds, whose smallest eigenvalue of the curvature operator of the second kind of $\mathring{R}$ satisfies $λ_1 \ge -θ(n) \barλ$, are either flat or round spheres, where $\bar λ$ is the average of the eigenvalues of $\mathring{R}$, and $θ(n)$ 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.