Complete manifolds with nonnegative curvature operator

In this short note, as a simple application of the strong result proved recently by Böhm and Wilking, we give a classification on closed manifolds with $2$-nonnegative curvature operator. Moreover, by the new invariant cone constructions of Böhm and Wilking, we show that any complete Riemannian manifold (with dimension $\ge 3$) whose curvature operator is bounded and satisfies the pinching condition $R\ge \delta \frac{\operatorname{tr}(R)}{2n(n-1)}\operatorname{I}>0$, for some $\delta>0$, must be compact. This provides an intrinsic analogue of a result of Hamilton on convex hypersurfaces.