Rigidity of Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind

We prove a rigidity theorem for closed Einstein manifolds of dimension $n\ge6$ under two lower bounds on the curvature operator of the second kind. More precisely, for $L_1, L_2\ge0$ and some real number $α>1$ in a suitable range, we consider \[ λ_1\ge-L_1\barλ, \qquad \frac1α\sum_{j=1}^αλ_j \ge-L_2