The total geodesic curvature and the $(2+1)$-dimensional hyperbolic mass

Abstract

In this paper, we derive a novel geometric inequality involving the total geodesic curvature, serving as an analogue of the Brown-York mass in the setting of $(2+1)$-dimensional gravity. The asymptotic behaviour of this newly defined quasi-local mass is examined for large ellipses in the time-symmetric slices of the Bañados-Teitelboim-Zanelli black hole solutions. Moreover, we establish an improved upper bound for the $(2+1)$-dimensional hyperbolic Bartnik mass.

Introduction rewritten; an improved upper bound estimate of the Bartnik mass added