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We consider a (2+1)-dimensional holographic CFT on a static spacetime withglobally timelike Killing vector. Taking the spatial geometry to be closed butotherwise general we expect a non-trivial vacuum energy at zero temperature dueto the Casimir effect. We assume a thermal state has an AdS/CFT dualdescription as a static smooth solution to gravity with a negative cosmologicalconstant, which ends only on the conformal boundary or horizons. A bulkgeometric argument then provides an upper bound on the ratio of CFT free energyto temperature. Considering the zero temperature limit of this bound impliesthe vacuum energy of the CFT is non-positive. Furthermore the vacuum energymust be negative unless the boundary metric is locally conformal to a productof time with a constant curvature space. We emphasise the argument does notrequire the zero temperature bulk geometry to be smooth, but only thatsingularities are `good' so are hidden by horizons at finite temperature.

Vacuum energy is non-positive for (2+1)-dimensional holographic CFTs