Barycenter technique for the higher order $Q$-curvature equation

TL;DR

This paper proves the existence of conformal metrics with constant higher order Q-curvature on certain manifolds using the barycenter method, without relying on positive mass theorems or sign conditions.

Contribution

It introduces a novel application of the barycenter technique to higher order Q-curvature equations, avoiding the need for positive mass assumptions.

Findings

Existence of conformal metrics with constant Q-curvature established.

Application of barycenter method to higher order geometric PDEs.

No requirement for positive mass condition on the GJMS operator.

Abstract

Let $k\ge1$ be an integer, and $(M,g)$ be a smooth, closed Riemannian manifold of dimension $2k+1\le n\le 2k+3$, or $(M,g)$ be locally conformally flat of dimension $n\ge 2k+1$. Applying the Bahri-Coron barycenter method, we show the existence of a conformal metric with constant $Q$-curvature of order $2k$, or equivalently, the existence of a positive solution for the $2k$-th order $Q$-curvature equation involving the GJMS operator $P_{g}$. We only assume a natural positivity preserving condition on $P_{g}$ and do not suppose any condition on the sign of the {\emph{mass}} of $P_{g}$. In particular, we obtain existence without using a positive mass theorem.