Abstract:
This is a joint work in progress with Wei Wei. In this talk we are discussing some new results in positive mass theorem. There are versions of  asymptotic hyperbolic manifolds where a proper mass term is defined for ends and proven to be positive when scalar curvature is negative definite. Corresponding Penrose type inequality is not yet known. We discussed that under the special case of conformally flat setting, when given a slightly stronger curvature condition ($\sigma_2$ curvature positive definite), the total mass is bounded below by geometric information from conic singularities. It is interesting to see if these singularities are related to back holes.