Geometry of three manifolds and existence of Black Hole due to boundary effect

In this paper, we observe that the brane functional studied in [5] can be used to generalize some of the works that Schoen and I [4] did many years ago. The key idea is that if a three dimensional manifold M has a boundary with strongly positive mean curvature, the effect of this mean curvature can influence the internal geometry of M . For example, if the scalar curvature of M is greater than certain constant related to this boundary effect, no incompressible surface of higher genus can exist. A remarkable statement in general relativity is that if the mean curvature of ∂M is strictly greater than the trace of pij (the second fundamental form of M in space time), the value of this difference can provide the existence of apparent horizon in M . In fact, matter density can even be allowed to be negative if this boundary effect is very strong.