A volume correspondence between anti-de Sitter space and its boundary

Abstract

Let H n+1 1 be the (n + 1)-dimensional anti-de Sitter space, in this paper we propose to extend the space by gluing it to another copy of H n+1 that is invariant under conformal symmetry, establishing a volume correspondence between DH n+1 1 and ∂H n+1 1 . 1 respectively) are consistent, in the sense that when DH n is treated as embedded in DH n+1 1 . Theorem 1.4. Let P ∈ H 0 in DH n+1 1, then V n+1 (P ) is real for n odd, and V n+1 (P ) is imaginary for n even and is completely determined by P ∩ ∂H n+1 1 .