Conformal Metrics with Constant Q-Curvature

TL;DR

This paper studies how to conformally modify metrics on four-dimensional manifolds to achieve constant Q-curvature, using a variational approach that involves the set of formal barycenters.

Contribution

It introduces a variational framework for the constant Q-curvature problem and connects it to the geometry of formal barycenters on the manifold.

Findings

Solutions are critical points of saddle type.

The problem naturally relates to the set of formal barycenters.

Provides a new perspective on the geometric analysis of Q-curvature.

Abstract

We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the manifold.