Comparison and Rigidity Theorems for geodesic curvatures in two dimensional Alexandrov spaces

In this work, we study geodesic curvature of the boundary of a two dimensional Alexandrov space of curvature bounded below (CBB). We prove several comparison and globalization theorems for the geodesic curvature, generalizing the known results for curves in space of curvature bounded above (CBA) by Alexander and Bishop (Differ. Geom. Appl. 6, No. 1, 67-86 (1996)). We also prove a rigidity theorem for boundary with corners and geodesic curvature lower bound. This generalizes the known rigidity result by Grove and Peterson (Geom. Topol. 26 (4) 1635 - 1668, (2022)) in 2d.