Manifolds with pinched 2-positive curvature operator (Q692645)

Let \((M,g)\) be a complete Riemannian manifold of dimension \(m\geq3\). Let \(Rm\) be the curvature operator from \(\Lambda^2(M)\) to \(\Lambda^2(M)\). This operator is said to satisfy the \(\delta\)-pinched two-positive condition if there exists \(\delta>0\) such that \[ \mu_1(Rm)+\mu_2(RM)\geq\delta\tau>0\,. \] Here \(\tau\) is the scalar curvature and \(\mu_1(Rm)\) and \(\mu_2(Rm)\) are the two smallest eigenvalues of \(Rm(g)\). The manifold \((M,g)\) is said to have positive (resp., nonnegative) complex sectional curvature if, given any two complex tangent vectors \(z\) and \(W\), one has \[ R(Z\wedge W,\bar Z\wedge\bar W)>0 \mathrm{, resp., }\geq0. \] The asymptotic volume ratio is defined as \[ AVR(g):=\lim_{R\rightarrow\infty}\left\{\text{Vol}(B_r(O))r^{-m}\right\}. \] This is independent of the base point \(O\) chosen. The authors prove the following two theorems. Theorem 1: Let \((M,g)\) be a complete Riemannian manifold with bounded and positive complex sectional curvature of dimension \(m\geq3\). If the curvature operator is \(\delta\)-pinched two-positive, then \(M\) is compact. Furthermore, the solution to the normalized Ricci flow with \(g(0)=g\) converges to a limit metric with constant positive sectional curvature. Theorem 2: Let \((M,g)\) be a complete Riemannian manifold with nonnegative (possibly unbounded) complex sectional curvature of dimension \(m\geq3\). If the curvature operator of \((M,g)\) is \(\delta\)-pinched \(2\)-positive and if \(AVR(g)>0\), then \(M\) must be compact.