Scalar curvature rigidity of the four-dimensional sphere

Abstract

Let $(M,g)$ be a closed connected oriented (possibly non-spin) smooth four-dimensional manifold with scalar curvature bounded below by $n(n-1)$. In this paper, we prove that if $f$ is a smooth map of non-zero degree from $(M,g)$ to the unit four-sphere, then $f$ is an isometry. Following ideas of Gromov, we use $\mu$-bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case of strict inequality. Our proof of rigidity is based on the harmonic map heat flow coupled with the Ricci flow.

Improved exposition. Comments are welcome!