About the mass of certain second order elliptic operators

Let (M,g)(M,g) be a closed Riemannian manifold of dimension n≥3n≥3 and let f∈C∞(M)f∈C∞(M), such that the operator Pf:=Δg+fPf:=Δg+f is positive. If g is flat near some point p and f vanishes around p  , we can define the mass of PfPf as the constant term in the expansion of the Green function of PfPf at p  . In this paper, we establish many results on the mass of such operators. In particular, if f:=n−24(n−1)sg, i.e. if PfPf 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.