Scalar curvature rigidity of the four-dimensional sphere

TL;DR

This paper proves that any smooth map of non-zero degree from a four-dimensional manifold with scalar curvature at least that of the sphere to the sphere must be an isometry, using advanced geometric flows and $mbda$-bubbles.

Contribution

It establishes a scalar curvature rigidity result for four-dimensional spheres, extending previous three-dimensional results using harmonic map heat flow and Ricci flow techniques.

Findings

Maps of non-zero degree are isometries under scalar curvature bounds.

Rigidity holds even for non-spin four-manifolds.

Uses harmonic map heat flow coupled with Ricci flow for proof.

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.