The Differentiable Sphere Theorem (After S. Brendle and R. Schoen)

Abstract

In these notes we describe a major result obtained recently using the Ricci flow technique in the context of positive curvature. It is due to S. Brendle and R. Schoen and states that a strictly 1/4-pinched closed manifold carries a metric of constant (positive) sectional curvature. It relies on a technique developed by C. Bohm and B. Wilking who obtained the same conclusion assuming that the manifold has positive curvature operator. The maximum principle applied to the Ricci flow equation leads to studying an ordinary differential equation on the space of curvature operators.