Epstein curves and holography of the Schwarzian action

Inspired by the duality between Jackiw-Teitelboim gravity and Schwarzian field theory, we show identities between the Schwarzian action of a circle diffeomorphism and (1) the length and the hyperbolic area enclosed by the Epstein curve in the hyperbolic disk $\mathbb{D}$, (2) the asymptotic excess in the isoperimetric inequality for the equidistant Epstein foliation, (3) the variation of the Loewner energy along its equipotential foliation, and (4) the asymptotic change in hyperbolic area under a conformal distortion near the circle. Together with the isoperimetric inequality and the monotonicity of the Loewner energy, our identities immediately imply two new proofs of the non-negativity of the Schwarzian action for circle diffeomorphisms. Moreover, we extend the identity (1) to the dual Epstein curve in de Sitter $2$-space, to the Schwarzian action for the coadjoint orbits $\mathrm{M"{o}b}_n (\mathbb{S}^1) \backslash \mathrm{Diff}(\mathbb{S}^1)$, and to piecewise M"obius circle diffeomorphisms which are only $C^{1,1}$ regular. We also show that the horocycle truncation used in the construction of the Epstein curve defines a renormalized length of hyperbolic geodesics in $\mathbb{D}$, which coincides with the log of the bi-local observable. From this, we show that the bi-local observables on the edges of any ideal triangulation of $\mathbb{D}$ determine the circle diffeomorphism.