# The Trace and the Mass of subcritical GJMS Operators

## TL;DR

This paper investigates the extremal properties of the zeta-regularized trace of the inverse of subcritical GJMS operators on compact manifolds, establishing bounds and conditions for maximizers within conformal classes.

## Contribution

It proves that the supremum of the trace over conformal metrics is at least that of the sphere and characterizes when the maximum is achieved by a constant mass metric.

## Key findings

- Supremum of the trace is always ≥ that of the sphere.
- Maximum is attained by a constant mass metric under certain conditions.
- Geometric conditions for the supremum to be strictly larger than the sphere's value.

## Abstract

Let $L_g$ be the subcritical GJMS operator on an even-dimensional compact manifold $(X, g)$ and consider the zeta-regularized trace $\mathrm{Tr}_\zeta(L_g^{-1})$ of its inverse. We show that if $\ker L_g = 0$, then the supremum of this quantity, taken over all metrics $g$ of fixed volume in the conformal class, is always greater than or equal to the corresponding quantity on the standard sphere. Moreover, we show that in the case that it is strictly larger, the supremum is attained by a metric of constant mass. Using positive mass theorems, we give some geometric conditions for this to happen.