Diameter estimate for closed manifolds with positive scalar curvature

TL;DR

This paper establishes an upper bound on the diameter of simply connected closed manifolds with positive scalar curvature, linking it to scalar curvature integrals, the Yamabe constant, and dimension, with sharpness demonstrated on spheres.

Contribution

It introduces a new diameter estimate for manifolds with positive scalar curvature, removing the Yamabe constant dependency under conformal immersion conditions.

Findings

Diameter bounds are sharp and achieved by round spheres.

The estimate depends on scalar curvature integral, Yamabe constant, and dimension.

Conformal immersion into spheres simplifies the diameter bound.

Abstract

For a simply connected closed Riemannian manifold with positive scalar curvature, we prove an upper diameter bound in terms of its scalar curvature integral, the Yamabe constant and the dimension of the manifold. When a manifold has a conformal immersion into a sphere, the dependency on the Yamabe constant is not necessary. The power of scalar curvature integral in these diameter estimates is sharp and it occurs at round spheres with canonical metric.