  In this paper, we observe that the brane functional studied in hep-th/9910245
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.
