Conformal metrics in ${\mathbb R}^{2m}$ with constant $Q$-curvature and arbitrary volume

TL;DR

This paper constructs radial solutions to a polyharmonic equation in even-dimensional Euclidean space, demonstrating the existence of conformal metrics with prescribed volume and constant positive Q-curvature, thus resolving open questions.

Contribution

It proves the existence of conformal metrics with arbitrary volume and positive constant Q-curvature in higher even dimensions, answering previously open questions.

Findings

Existence of radial solutions with prescribed volume for ^u in \\mathbb{R}^{2m}

Construction of conformal metrics with positive constant Q-curvature and arbitrary volume

Resolution of open problems posed by Martinazzi

Abstract

We study the polyharmonic problem $\Delta^m u = \pm e^u$ in ${\mathbb R}^{2m}$, with $m \geq 2$. In particular, we prove that {\sl for any} $V > 0$, there exist radial solutions of $\Delta^m u = -e^u$ such that $$\int_{{\mathbb R}^{2m}} e^u dx = V.$$ It implies that for $m$ odd, given arbitrary volume $V > 0$, there exist conformal metrics $g$ on ${\mathbb R}^{2m}$ with positive constant $Q$-curvature and vol$(g) =V$. This answers some open questions in Martinazzi's work.