The authors prove a positive mass theorem in the weighted setting. More specifically, a weighted manifold is a Riemannian manifold \((M, g)\) endowed with a function \(f: M \rightarrow\) \(\mathbb{R}\), defining the measure \(e^{-f} d V_g\). The weighted mass of an Asymptotically Euclidean (AE) manifold with weight function \(f\) is defined as \[ \mathrm{m}_f(g):=\mathfrak{m}(g)+2 \lim _{\rho \rightarrow \infty} \int_{S_\rho}\langle\nabla f, v\rangle e^{-f} d A, \] where \(S_\rho\) is a coordinate sphere of radius \(\rho\) with outward normal \(\nu\) and area form \(d A\), \(\mathfrak{m}(g)\) is the usual ADM-mass.
Then the authors prove that when \((M,g)\) is AE and spin, one has the weighted Witten equation: \[ \mathrm{m}_f(g)=4 \int_M\left(|\nabla \psi|^2+\frac{1}{4} \mathrm{R}_f|\psi|^2\right) e^{-f} d V_g, \] where \(\mathrm{R}_f\) is the weighted scalar curvature and \(\psi\) is a weighted harmonic spinor that is asymptotically constant near the infinity. Hence, they prove the weighted positive mass theorem if \(\mathrm{R}_f\geq0\). Additionally, the rigidity result is derived.
Moreover, the author shows the monotonicity of the weighted mass under Ricci flow.