Rigidity of noncompact complete manifolds with harmonic curvature

TL;DR

This paper proves that noncompact complete manifolds with harmonic curvature are Einstein under certain smallness conditions on curvature norms, extending rigidity results in geometric analysis.

Contribution

It establishes new conditions under which noncompact harmonic curvature manifolds are necessarily Einstein, particularly involving $L_{n/2}$ norm smallness for higher dimensions.

Findings

Manifolds are Einstein if $L_{n/2}$ norms are sufficiently small.

Results apply to manifolds with positive Sobolev constant and finite curvature norms.

Extension of rigidity theorems to noncompact harmonic curvature manifolds.

Abstract

Let $(M,g)$ be a noncompact complete $n$-manifold with harmonic curvature and positive Sobolev constant. Assume that $L_2$ norms of Weyl curvature and traceless Ricci curvature are finite. We prove that $(M,g)$ is Einstein if $n \ge 5$ and $L_{n/2}$ norms of Weyl curvature and traceless Ricci curvature are small enough.