For a path in a compact finite dimensional Alexandrov space X with curv  , the two basic geometric invariants are the length and the turning angle (which measures the closeness from being a geodesic). We show that the sum of the two invariants of any loop is bounded from below in terms of , the dimension, diameter, and Hausdorff measure of X. This generalizes a basic estimate of Cheeger on the length of a closed geodesic in a closed Riemannian manifold ([Ch], [GP1,2]). To see that the above result also generalizes and improves an analog of the Cheeger type estimate in Alexandrov geometry in [BGP], we show that for a class of subsets of X, the n-dimensional Hausdorff measure and rough volume are proportional by a constant depending on n = dim(X).

In this addendum we complete the proof of Theorem 7.10 in [1].

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We study weakly trapped submanifolds of codimension two in a standard static spacetime. In this setting, we apply some generalized maximum principles in order to investigate the geometry of these trapped submanifolds. For instance, we establish sufficient conditions to guarantee that such a spacelike submanifold must be a hypersurface of the Riemannian base of the ambient spacetime. As a consequence, we prove that there do not exist n-dimensional compact (without boundary) trapped submanifolds immersed in an (n+2)-dimensional standard static spacetime. Such a nonexistence result was originally obtained for stationary spacetimes by Mars and Senovilla [20]. Furthermore, we also investigate parabolic weakly trapped submanifolds immersed in a standard static spacetime.

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.