About the mass of certain second order elliptic operators

Let $(M,g)$ be a closed Riemannian manifold of dimension $n \geq 3$ and let $f\in C^{\infty}(M)$, such that the operator $P_f:= \Delta_g+f$ is positive. If $g$ is flat near some point $p$ and $f$ vanishes around $p$, we can define the mass of $P_f$ as the constant term in the expansion of the Green function of $P_f$ at $p$. In this paper, we establish many results on the mass of such operators. In particular, if $f:= \frac{n-2}{4(n-1)} \scal_g$, i.e. if $P_f$ is the Yamabe operator, we show the following result: assume that there exists a closed simply connected non-spin manifold $M$ such that the mass is non-negative for every metric $g$ as above on $M$, then the mass is non-negative for every such metric on every closed manifold of the same dimension as $M$.