Conformal metrics on R^{2m} with constant Q-curvature and large volume

TL;DR

This paper investigates conformal metrics with constant Q-curvature on Euclidean spaces, revealing existence results for large or small volumes depending on the dimension and curvature sign, contrasting with known four-dimensional cases.

Contribution

It extends known existence results of conformal metrics with prescribed Q-curvature to higher dimensions and different curvature signs, highlighting new volume thresholds.

Findings

Existence of conformal metrics with prescribed volume for large V in 6D.

Existence of conformal metrics with prescribed volume for small V in odd dimensions > 1.

Results differ significantly from the four-dimensional case.

Abstract

We study conformal metrics on R^{2m} with constant Q-curvature and finite volume. When m=3 we show that there exists V* such that for any V\in [V*,\infty) there is a conformal metric g on R^{6} with Q_g = Q-curvature of S^6, and vol(g)=V. This is in sharp contrast with the four-dimensional case, treated by C-S. Lin. We also prove that when $m$ is odd and greater than 1, there is a constant V_m>\vol (S^{2m}) such that for every V\in (0,V_m] there is a conformal metric g on R^{2m} with Q_g = Q-curvature of S^{2m}, vol(g)=V. This extends a result of A. Chang and W-X. Chen. When m is even we prove a similar result for conformal metrics of negative Q-curvature.