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We discuss the possible relevance of complex codimension-two extremalsurfaces to the the Ryu-Takayanagi holographic entanglement proposal and itscovariant Hubeny-Rangamani-Takayanagi (HRT) generalization. Such surfaces livein a complexified bulk spacetime defined by analytic continuation. We identifysurfaces of this type for BTZ, Schwarzschild-AdS, and Schwarzschild-Lifshitzplanar black holes. Since the dual CFT interpretation for the imaginary part oftheir areas is unclear, we focus on a straw man proposal relating CFT entropyto the real part of the area alone. For Schwarzschild-AdS andSchwarzschild-Lifshitz, we identify families where the real part of the areaagrees with qualitative physical expectations for the appropriate CFT entropyand, in addition, where it is smaller than the area of corresponding realextremal surfaces. It is thus plausible that the CFT entropy is controlled bythese complex extremal surfaces.

Complex Entangling Surfaces for AdS and Lifshitz Black Holes?