Critical metrics of the $L^2$-norm of the scalar curvature

TL;DR

This paper studies complete critical metrics related to the $L^2$-norm of scalar curvature, proving constant scalar curvature for positive cases and characterizing nonnegative cases in dimensions three and four.

Contribution

It establishes that complete critical metrics with positive scalar curvature have constant scalar curvature and characterizes nonnegative scalar curvature cases in 3D and 4D.

Findings

Complete critical metrics with positive scalar curvature have constant scalar curvature.

Characterization of critical metrics with nonnegative scalar curvature in dimensions three and four.

Abstract

In this paper we investigate complete critical metrics of the $L^{2}$-norm of the scalar curvature. We prove that any complete critical metric with positive scalar curvature has constant scalar curvature and we characterize critical metrics with nonnegative scalar curvature in dimension three and four.