A new framework for solving the hierarchy problem was recently proposed which does not rely on low energy supersymmetry or technicolor. The fundamental Planck mass is at a TeV and the observed weakness of gravity at long distances is due the existence of new sub-millimeter spatial dimensions. In this picture the standard model fields are localized to a (3+ 1)-dimensional wall or '3-brane'. The hierarchy problem becomes isomorphic to the problem of the largeness of the extra dimensions. This is in turn inextricably linked to the cosmological constant problem, suggesting the possibility of a common solution. The radii of the extra dimensions must be prevented from both expanding to too great a size, and collapsing to the fundamental Planck length TeV(sup -1). In this paper we propose a number of mechanisms addressing this question. We argue that a positive bulk cosmological constant anti-Lambda can stabilize the internal manifold against expansion, and that the value of anti-Lambda is not unstable to radiative corrections provided that the supersymmetries of string theory are broken by dynamics on our 3-brane.
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