Bach-pinched metrics on closed manifolds

TL;DR

This paper demonstrates the existence of specific metrics on all closed four-dimensional manifolds where the Bach tensor is proportionally controlled by the scalar curvature, extending deformation techniques in geometric analysis.

Contribution

It introduces a new application of Aubin's deformation method to construct metrics with Bach tensor pinched by scalar curvature on all closed four-manifolds.

Findings

Existence of metrics with Bach tensor pinched by scalar curvature on all closed four-manifolds

Extension of deformation techniques to control Bach tensor in four dimensions

New insights into geometric structures via scalar curvature and Bach tensor relations

Abstract

Exploiting the deformation method introduced by Aubin in his seminal work to construct constant negative scalar curvature metrics, we show the existence, on every closed manifold of dimension four, of a metric whose Bach tensor is pinched by the scalar curvature.