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In this paper, we observe that the brane functional studied in hep-th/9910245can be used to generalize some of the works that Schoen and I [4] did manyyears ago. The key idea is that if a three dimensional manifold M has aboundary with strongly positive mean curvature, the effect of this meancurvature can influence the internal geometry of M. For example, if the scalarcurvature of M is greater than certain constant related to this boundaryeffect, no incompressible surface of higher genus can exist.

Geometry of three manifolds and existence of black hole due to boundary effect