A variational characterization of flat spaces in dimension three

TL;DR

This paper proves that in three dimensions, the only complete metrics with non-negative scalar curvature that are critical points of the -curvature functional are flat metrics, providing a variational characterization.

Contribution

It establishes a unique variational characterization of flat spaces in three dimensions through the -curvature functional.

Findings

Flat metrics are the only complete, non-negatively curved critical points in three dimensions.

The result characterizes flat spaces via a variational principle involving -curvature.

Provides a new perspective on the geometry of three-dimensional manifolds.

Abstract

In this short note we prove that, in dimension three, flat metrics are the only complete metrics with non-negative scalar curvature which are critical for the $\sigma_{2}$-curvature functional.