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The stress tensor is a basic local operator in any field theory; in thecontext of AdS/CFT, it is the operator which is dual to the bulk geometryitself. Here we exploit this feature by using the bulk geometry to placeconstraints on the local energy density in static states of holographic$(2+1)$-dimensional CFTs living on a closed (but otherwise generally curved)spatial geometry. We allow for the presence of a marginal scalar deformation,dual to a massless scalar field in the bulk. For certain vacuum states in whichthe bulk geometry is well-behaved at zero temperature, we find that the bulkequations of motion imply that the local energy density integrated overspecific boundary domains is negative. In the absence of scalar deformations,we use the inverse mean curvature flow to show that if the CFT spatial geometryhas spherical topology but non-constant curvature, the local energy densitymust be positive somewhere. This result extends to other topologies, but onlyfor certain types of vacuum; in particular, for a generic toroidal boundary,the vacuum's bulk dual must be the zero-temperature limit of a toroidal blackhole.

Bounds on the local energy density of holographic CFTs from bulk geometry