A non-spin method to the positive weighted mass theorem for weighted manifolds (Q6572138)

A weighted manifold (also called manifold with density) is a smooth Riemannian manifold \((M, g)\) endowed with a weighted measure \(e^{-f}\mathrm{dvol}_g\), where \(f\) is a smooth function on \(M\) and \(\mathrm{dvol}_g\) is the volume element of \((M, g)\). This leads to the notion of weighted scalar curvature (or \(P\)-scalar curvature), defined as\N\[\NP(f,g)=R(g)+2\Delta_{g}f-|df|^{2}_{g}.\N\]\NWeighted manifolds with positive or non-negative \(P\)-scalar curvature have very similar properties compared to Riemannian manifolds with positive or non-negative scalar curvature.\N\NA weighted manifold \((M^n , g, f)\) is said to be asymptotically flat (AF) of \(( p,\tau )\)-type if \(n \geq 3\) and there is a compact subset \(K\subset M\) such that:\N\begin{itemize}\N\item[1.] The complement \(M \setminus K\) consists of finitely many ends \(E_1, E_2, \dots, E_ N\) and each \(E_k\) is diffeomorphic to \(\mathbb{R}^n\) minus the unit ball \(B_1\);\N\item[2.] In the above coordinate chart, on each \(E_{k}\) the components \(g_{ij}-\delta_{i j}\) and the function \(f\) belong to \(W^{2,p}_{-\tau}(E_k)\);\N\item[3.] The scalar curvature \(R(g)\) and the Laplacian \(\Delta_{g}f\) belong to \(L^{\infty}_{-2\tau-2}\)\N\item[4.] The \(P\)-scalar curvature \(P(g, f )\) belongs to \(L^1\) ;\N\item[5.] When \(n = 3\), the functions \(g_{ij}-\delta_{ij}\) belong to \(W^{1,\infty}_{-1}(E_{k})\) on each end \(E_{k}\).\N\end{itemize}\N\NGiven an AF weighted manifold \((M^n , g, f)\) of \(( p,\tau )\)-type, the weighted mass of an end \(E\) is defined to be\N\[\Nm(M, g, f , E) = \lim_{\rho \to +\infty}\int_{S_{\rho}}(\partial_{j} g_{ij} -\partial_{i} g_{jj} + 2\partial_{i} f ) \nu^i d\sigma,\N\]\Nwhere \(S_{\rho}\) is the sphere of radius \(\rho\) in \(\mathbb{R}^{n}\) centered at the origin, \(\nu\) is its outward-pointing normal and \(d\sigma\) is its area form.\N\NThe authors prove a positive mass theorem for AF weighted manifolds.\N\NTheorem. Let \(3\leq n\leq 7\). If \((M^n , g, f )\) is an AF weighted manifold of \(( p, \tau)\)-type with non-negative \(P\)-scalar curvature, then for each end \(E\) we have \(m(M, g, f , E)\geq 0\). Moreover, if the equality holds for some end \(E\), then \((M, g)\) is isometric to the Euclidean \(n\)-space and \(f\) is the zero function.