Rigidity of noncompact complete Bach-flat manifolds

TL;DR

This paper establishes conditions under which noncompact complete Bach-flat manifolds are flat or conformally flat, based on scalar curvature and small curvature bounds, advancing understanding of their geometric rigidity.

Contribution

It provides new rigidity results for Bach-flat manifolds with positive Yamabe constant, characterizing flatness or conformal flatness under small curvature bounds and scalar curvature conditions.

Findings

Flat if scalar curvature is zero and curvature bound is small

Conformal to flat space if scalar curvature is nonconstant and bounds are small

Extends rigidity results for Bach-flat manifolds with curvature constraints

Abstract

Let $(M,g)$ be a noncompact complete Bach-flat manifold with positive Yamabe constant. We prove that $(M,g)$ is flat if $(M, g)$ has zero scalar curvature and sufficiently small $L_{2}$ bound of curvature tensor. When $(M, g)$ has nonconstant scalar curvature, we prove that $(M, g)$ is conformal to the flat space if $(M, g)$ has sufficiently small $L_2$ bound of curvature tensor and $L_{4/3}$ bound of scalar curvature.