*(English)*Zbl 0978.53085

Let $M$ be an $(n+1)$-dimensional manifold with non-empty boundary $N$. A metric $g$ on $M$ is conformally compact if it can be written as $h/{r}^{2}$, where $h$ is a compact metric on $M$, and $r$ is a function on $M$ with a first-order zero on the boundary. $(M,g)$ 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 $M$ with a compact manifold, the conformal field theory (CFT) on $N$ should be related to the string theory in $M$. More precisely, if $c$ is a conformal class on $N$, the partition function of the CFT on $(N,c)$ should be a sum of terms corresponding to the string theory on the $({M}_{i},{g}_{i})$, where $\partial {M}_{i}=N$ and the ${g}_{i}$ are Einstein conformally compact metrics on ${M}_{i}$ inducing $c$ on $N$.

The CFT on $N$ makes sense mostly when $(N,c)$ has positive scalar curvature, in the sense that there is a metric on $N$ conformal to $c$ which has positive scalar curvature. The main result of the paper is that if, in addition, $\partial M=N$ and $g$ is an Einstein, conformally compact metric on $M$ inducing $c$ on $N$, then ${H}_{n}(M,Z)=0$.

The authors point out that this implies that $N$ is connected, and also that $M$ contains no “wormhole”; it therefore removes some puzzling questions concerning the AdS-CFT correspondence.

To prove the result, the authors introduce the “brane action” $L$ on embedded hypersurfaces of $M$, defined as the area minus $n$ times a primitive of the volume form. Then they prove using a local computation that $L$ has no minimum, while the behavior of the function at infinity shows that there is one in each non-zero class of ${H}_{n}(M,Z)$.

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).

##### MSC:

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

53C20 | Global Riemannian geometry, including pinching |

53C80 | Applications of global differential geometry to physics |

81T30 | String and superstring theories |

83E30 | String and superstring theories |