Embedded minimal tori in S^3 and the Lawson conjecture

We show that any embedded minimal torus in S^3 is congruent to the Clifford torus. This answers a question posed by H.B. Lawson, Jr., in 1970.Comment: Final version, to appear in Acta Mathematic

Einstein manifolds with nonnegative isotropic curvature are locally symmetric

Let (M,g) be an Einstein manifold of dimension n \geq 4 with nonnegative isotropic curvature. We show that (M,g) is locally symmetric.Comment: Final versio

A general convergence result for the Ricci flow in higher dimensions

Let (M,g_0) be a compact Riemannian manifold of dimension n \geq 4. We show that the normalized Ricci flow deforms g_0 to a constant curvature metric provided that (M,g_0) x R has positive isotropic curvature. This condition is stronger than 2-positive flag curvature but weaker than 2-positive curvature operator.Comment: Final version, to appear in Duke Math Journa