Lorentz--Epstein surfaces and a Liouville action for positive curves

TL;DR

This paper explores the analogs of Epstein surfaces and Liouville action within the context of anti-de Sitter space and conformal metrics, introducing invariants for positive curves in flag manifolds.

Contribution

It defines new geometric invariants for positive curves in flag manifolds using analogs of Epstein surfaces and Liouville action in the AdS/CFT framework.

Findings

Defined Epstein surfaces and Liouville action analogs in AdS space.

Constructed invariants for positive curves in flag manifolds.

Invariants are finite for piecewise circular curves.

Abstract

We investigate and define in this paper, in the context of the correspondence between anti-de Sitter $3$-space and $(1,1)$-conformal metrics, the analogs of $\cW$-volume, Epstein surfaces, and Liouville action. These notions were well-studied in the correspondence between $3d$-hyperbolic manifolds and $2d$ conformal metrics. We apply our construction to positive curves in flag manifolds equipped with a positive structure to obtain invariants of these curves that are finite in the case of piecewise circles.