Q-curvature Flow for GJMS Operators with Non-trivial Kernel

TL;DR

This paper studies the behavior of a geometric flow related to GJMS operators on compact manifolds, establishing conditions for global existence and convergence when total Q-curvature is negative, and finite-time blow-up when positive.

Contribution

It introduces a conformally invariant condition on kernel nodal domains that guarantees flow convergence for negative total Q-curvature.

Findings

Flow converges to a prescribed Q-curvature metric when total Q-curvature is negative.

Flow blows up in finite time when total Q-curvature is positive.

Identifies a new invariant condition related to kernel nodal domains.

Abstract

We investigate the prescribed Q-curvature flow for GJMS operators with non-trivial kernel on compact manifolds of even dimension. When the total Q-curvature is negative, we identify a conformally invariant condition on the nodal domains of functions in the kernel of the GJMS operator, allowing us to prove the global existence and the convergence of the flow to a metric which is conformal to the initial one, and having a prescribed Q-curvature. If the total Q-curvature is positive, we show that the flow blows up in finite time.