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

We consider a Jordan domain diffeomorphic to a closed two-dimensional disk with a smooth boundary. Assuming the Gauss curvature of the domain has a negative lower bound, the Gauss-Bonnet formula provides an upper bound for the total geodesic curvature of the boundary curve. This bound, however, inherently depends on the interior geometry of the region. In this paper, we derive an upper bound for the total geodesic curvature expressed solely in terms of the boundary data. Notably, the proof is connected to the positivity of the hyperbolic Hamiltonian mass in the (2+1)-dimensional gravity theory.