 Solution

On the $t=0$ slice, rotational symmetry lets the surface be written $z=z(r)$. The inversion symmetry fixing the boundary circle extends to an isometry of the Poincare half-space whose fixed set isThe unique extremal surface anchored at $r=R,z=0$ must be invariant under that isometry. Hence the Ryu–Takayanagi surface for a disk is the hemisphereOne may verify directly that it extremizes the area functional $2\pi \int r{z}^{-2}\sqrt{1+z^{\prime 2}}dr$.