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

45 pages, 3 figures.This second version of the paper introduces an equivariant case. We have also restructured it to clearly define and separate W-volume and Liouville action. Finally, we have added the torus case