de Sitter extremal surfaces

We study extremal surfaces in de Sitter space in the Poincare slicing in the upper patch, anchored on spatial subregions at the future boundary ${\cal I}^+$, restricted to constant boundary Euclidean time slices (focussing on strip subregions). We find real extremal surfaces of minimal area as the boundaries of past lightcone wedges of the subregions in question: these are null surfaces with vanishing area. We also find complex extremal surfaces as complex extrema of the area functional, and the area is not always real-valued. In $dS_4$ the area is real. The area has structural resemblance with entanglement entropy in a dual $CFT$. There are parallels with analytic continuation from the Ryu-Takayanagi expressions for holographic entanglement entropy in $AdS$. We also discuss extremal surfaces in the $dS$ black brane and the de Sitter "bluewall" studied previously. The $dS_4$ black brane complex surfaces exhibit a real finite cutoff-independent extensive piece. In the bluewall geometry, there are real surfaces that go from one asymptotic universe to the other through the Cauchy horizons.

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