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

TL;DR

This paper investigates geodesic curvature in two-dimensional Alexandrov spaces with curvature bounded below, establishing comparison, globalization, and rigidity theorems that extend previous results from curvature bounded above.

Contribution

It generalizes known comparison and rigidity theorems for geodesic curvature from spaces with curvature bounded above to those with curvature bounded below in 2D Alexandrov spaces.

Findings

Proved comparison theorems for geodesic curvature in CBB spaces.

Established globalization theorems extending local results.

Derived a rigidity theorem for boundary with corners and curvature bounds.

Abstract

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.