A formal Riemannian structure on conformal classes and uniqueness for the $\sigma_{2}$-Yamabe problem

TL;DR

This paper introduces a new Riemannian metric on conformal classes of four-manifolds related to the $\sigma_2$-Yamabe problem, establishing solution uniqueness except for the sphere case.

Contribution

It defines a novel formal Riemannian structure on conformal classes and uses it to prove uniqueness of solutions to the $\sigma_2$-Yamabe problem.

Findings

Solutions are unique unless the manifold is conformally equivalent to the round sphere.

A new variational structure is established for the $\sigma_2$-Yamabe problem.

The formal Riemannian metric aids in analyzing solution properties.

Abstract

We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the $\sigma_2$-Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.