Higher order Bol's inequality and its applications

TL;DR

This paper establishes volume comparison and rigidity theorems using Q-curvature in conformal geometry, and applies these results to solve an open problem on conformally invariant equations.

Contribution

It introduces new volume comparison and rigidity theorems involving Q-curvature and applies them to characterize solutions of conformally invariant equations.

Findings

Volume comparison theorems in conformal classes using Q-curvature

Volume rigidity results for four-dimensional manifolds with non-negative scalar curvature

Conditions for existence of solutions to conformally invariant equations

Abstract

In the conformal class of Euclidean space, we give some volume comparison theorems with help of Q-curvature. Meanwhile, for compact four dimensional manifolds with non-negative scalar curvature, we give a volume rigidity theorem with respect to Q-curvature. Finally, we make use of these results to give some sufficient and necessary conditions for the existence of solutions to some conformally invariant equations which answers an open problem raised by Hyder-Martinazzi (2021, JDE).