The case of equality for the spacetime positive mass theorem

The rigidity of the spacetime positive mass theorem states that an initial data set $(M,g,k)$ satisfying the dominant energy condition with vanishing mass can be isometrically embedded into Minkowski space. This has been established by Beig-Chru\'sciel and Huang-Lee under additional decay assumptions for the energy and momentum densities $\mu$ and $J$. In this note we give a new and elementary proof in dimension 3 which removes these additional decay assumptions. Our argument uses spacetime harmonic functions and Liouville's theorem. We also provide an alternative proof based on the Killing development of $(M,g,k)$.Comment: Comments welcome

Stability of Llarull's theorem in all dimensions

We prove stability of Llarull's theorem in all dimensions using spin geometry. Our results are stated in terms of both intrinsic flat convergence and $C^0$ convergence outside a small set.Comment: Comments welcom

A Photocatalytic Organic Wastewater Disposing System Based on Solar Energy

AbstractThe paper presents a photocatalytic organic wastewater disposing system, with functions of automatic examination and drainage, for disposing organic wastewater. It is based on photocatalysis and oxidization of complex TiO2 nonmaterial, the system is made up of complex TiO2 nonmaterial carrier units, examining and controlling unit, solar energy unit and auxiliary magnetic field unit. Light absorption diagram shows the vanishing of impurity after adequate time