Higher order Bol's inequality and its applications

Mingxiang Li, Juncheng Wei

Published 2023-08-22Version 1

Assuming that the conformal metric $g=e^{2u}|dx|^2$ on $\mathbb{R}^n$ is normal, our focus lies in investigating the following conjecture: if the Q-curvature of such a manifold is bounded from above by $(n-1)!$, then the volume is sharply bounded from below by the volume of the standard n-sphere. In specific instances, such as when $u$ is radially symmetric or when the Q-curvature is represented by a polynomial, we provide a positive response to this conjecture, although the general case remains unresolved. Intriguingly, under the normal and radially symmetric assumptions, we establish that the volume is bounded from above by the volume of the standard n-sphere when the Q-curvature is bounded from below by $(n-1)!$, thereby addressing a certain open problem raised by Hyder-Martinazzi (2021, JDE).