Let be an -dimensional manifold with non-empty boundary . A metric on is conformally compact if it can be written as , where is a compact metric on , and is a function on with a first-order zero on the boundary. is then complete, and its boundary has a well-defined conformal structure.
The AdS-CFT correspondence is an important conjecture which states that, maybe after taking the tensor product of with a compact manifold, the conformal field theory (CFT) on should be related to the string theory in . More precisely, if is a conformal class on , the partition function of the CFT on should be a sum of terms corresponding to the string theory on the , where and the are Einstein conformally compact metrics on inducing on .
The CFT on makes sense mostly when has positive scalar curvature, in the sense that there is a metric on conformal to which has positive scalar curvature. The main result of the paper is that if, in addition, and is an Einstein, conformally compact metric on inducing on , then .
The authors point out that this implies that is connected, and also that contains no “wormhole”; it therefore removes some puzzling questions concerning the AdS-CFT correspondence.
To prove the result, the authors introduce the “brane action” on embedded hypersurfaces of , defined as the area minus times a primitive of the volume form. Then they prove using a local computation that has no minimum, while the behavior of the function at infinity shows that there is one in each non-zero class of .
Since this paper appeared, several alternate proofs of its main result have been provided, in particular by M. T. Anderson [Adv. Math. 179, No.2, 205-249 (2003; Zbl 1048.53032); see also math.DG/0104171], by M. Cai and G. J. Galloway [Adv. Theor. Math. Phys. 3, No. 6, 1769-1783 (2000; Zbl 0978.53084)] and by X. Wang (preprint).