Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2000-03-07
Adv.Theor.Math.Phys. 3 (1999) 1769-1783
Physics
High Energy Physics
High Energy Physics - Theory
12 pages, Latex2e
Scientific paper
In hep-th/9910245, Witten and Yau consider the AdS/CFT correspondence in the context of a Riemannian Einstein manifold $M^{n+1}$ of negative Ricci curvature which admits a conformal compactification with conformal boundary $N^n$. They prove that if the conformal class of the boundary contains a metric of positive scalar curvature, then $M$ and $N$ have several desirable properties: (1) $N$ is connected, (2) the $n$th homology of the compactified $M$ vanishes, and (3) the fundamental group of $M$ is "bounded by" that of $N$. Here it is shown that all of these results extend to the case where the conformal class of the boundary contains a metric of nonnegative scalar curvature. (The case of zero scalar curvature is of interest as it is borderline for the stability of the theory.) The proof method used here is different from, and in some sense dual to, that used by Witten and Yau. While their method involves minimizing the co-dimension one brane action on $M$, and requires the machinery of geometric measure theory, the main arguments presented here use only geodesic geometry.
Cai Mingliang
Galloway Gregory J.
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