Manifolds with nonnegative isotropic curvature

Abstract

We prove that if $(M^n,g)$, $n \ge 4$, is a compact, orientable, locally irreducible Riemannian manifold with nonnegative isotropic curvature, then one of the following possibilities hold: (i) $M$ admits a metric with positive isotropic curvature (ii) $(M,g)$ is isometric to a locally symmetric space (iii) $(M,g)$ is K\"ahler and biholomorphic to $\C P^\frac {n}{2}$. (iv) $(M,g)$ is quaternionic-K\"ahler. This is implied by the following result: Let $(M^{2n},g)$ be a compact, locally irreducible K\"ahler manifold with nonnegative isotropic curvature. Then either $M$ is biholomorphic to $\C P^n$ or isometric to a compact Hermitian symmetric space. This answers a question of Micallef and Wang in the affirmative. The proof is based on the recent work of S. Brendle and R. Schoen on the Ricci flow.