g_{\mu \nu }=\tilde{g}_{\mu \nu }+h_{\mu \nu }
and fall-off conditions w.r.t r, where \tilde{g}_{\mu \nu } is the AdS soliton metric. Horowitz and Myers go on to propose two conjectures, which are as follows:
- SUGRA solutions: g_{\mu \nu }=\tilde{g}_{\mu \nu } is the only solution to D=10 type IIB SUGRA, taking \tilde{g}_{\mu \nu }\times S^{5}_{l} for which \mathcal{E}= 0, and \mathcal{E}\geq 0 in general.
- Geometric solutions: g_{\mu \nu }=\tilde{g}_{\mu \nu } is the only solution to D=5 field equations with \Lambda = -6/l^{2} so that \mathcal{E}=0, and \mathcal{E}\geq 0 in general.
Seemingly, these are still open problems. However, I believe that resolving at least some aspects of these conjectures will go a long way in making sense of the geometric description of AdS/CFT.
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