Proof of the positive mass conjecture (Q859639)

Let \((V_n, g)\), \(n>2\), be a compact Riemannian manifold. If the conformal Laplacian \(L\) is invertible, the author shows under necessary hypotheses, that if the Green function \(G_L\) of \(L\) is of the form \(G_L(P, Q)=\frac{1}{n-2}\omega_{n-1}r^{n-2}+H(P, Q)\), where \(r=d(P,Q)\), with \(H(P, Q)\) bounded on \(V\), then \[ \lim_{r\to 0}r^{1-n}\int_{\partial B_P(r)}H(P,Q)\,d\sigma(Q)>0. \] From this theorem the author deduces theorems on the positive mass and on the \(C^0\)-compactness of the set of the solutions of the Yamabe equation.