On locally conformally flat manifolds with finite total $Q$-curvature

TL;DR

This paper investigates the geometric properties of locally conformally flat complete manifolds with finite total $Q$-curvature, establishing a quantization result linking total $Q$-curvature to a universal constant, thus highlighting its controlling role in geometry.

Contribution

It proves that the total $Q$-curvature of such manifolds is quantized as an integral multiple of a universal constant, extending understanding of $Q$-curvature's geometric significance.

Findings

Total $Q$-curvature equals an integer multiple of a universal constant.

$Q$-curvature controls the geometry similarly to Gaussian curvature in 2D.

Provides evidence of $Q$-curvature's fundamental role in higher-dimensional conformal geometry.

Abstract

In this paper, we focus our study on the ends of a locally conformally flat complete manifold with finite total $Q$-curvature. We prove that for such a manifold, the integral of the $Q$-curvature equals an integral multiple of a dimensional constant $c_n$, where $c_n$ is the integral of the $Q$-curvature on the unit $n$-sphere. It provides further evidence that the $Q$-curvature on a locally conformally flat manifold controls geometry as the Gaussian curvature does in two dimension.