The positive mass theorem with arbitrary ends (Q6593664)

The Riemannian positive mass theorem is a landmark result in geometric analysis and the study of scalar curvature. It states that in a complete Riemannian \(3\)-manifold \((M,g)\) with finitely many ends that are all asymptotically Schwarzschild, the mass of each end is nonnegative. This theorem was generalized in many ways, for example:\N\begin{itemize}\N\item to the asymptotically flat setting of each end;\N\item to dimensions \(\leq 7\) by \textit{R. M. Schoen} [Lect. Notes Math. None, 120--154 (1989; Zbl 0702.49038)] and to all dimensions by \textit{R. Schoen} and \textit{S.-T. Yau} [Surv. Differ. Geom. 24, 441--480 (2022; Zbl 07817751)];\N\item to initial data sets.\N\end{itemize}\NThis paper provides a further generalization of the positive mass theorem to the setting of manifolds with a distinguished asymptotically Schwarzschild end and otherwise arbitrary and possibly incomplete other ends. By work of \textit{R. Schoen} and \textit{S.-T. Yau} [Lectures on differential geometry. Cambridge, MA: International Press (1994; Zbl 0830.53001)], this implies the following Liouville-type theorem:\N\NTheorem. Let \((M,g)\) be a complete, locally conformally flat \(n\)-manifold \((n\geq 3)\) with nonnegative scalar curvature. If \(\Phi:M \rightarrow S^{n}\) is a conformal map, then \(\Phi\) is injective and \(\partial \Phi(M)\) has zero Newtonian capacity.