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Summary: L 2 CURVATURE AND VOLUME RENORMALIZATION OF AHE METRICS
ON 4-MANIFOLDS
MICHAEL T. ANDERSON
Abstract. This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds
M with the renormalized volume V , as de ned in the AdS/CFT correspondence, for asymptotically
hyperbolic Einstein metrics on M . In addition we compute and discuss the di erential or variation
dV of V , or equivalently the variation of the L 2 norm of the Weyl curvature, on the space of such
Einstein metrics.
0. Introduction.
The Chern-Gauss-Bonnet formula for a compact Riemannian 4-manifold (M; g) without bound-
ary states that
1
8 2
Z
M
(jRj 2 4jzj 2 )dV = 1
8 2
Z
M
(jW j 2 1
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