The Trace and the Mass of subcritical GJMS Operators

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.