A quantitative Schur comparison theorem for curves in CAT(k) spaces

Mohammad Ghomi; John Ioannis Stavroulakis

A quantitative Schur comparison theorem for curves in CAT(k) spaces

Differential Geometry 2026-07-16 v1 Metric Geometry

Authors: Mohammad Ghomi , John Ioannis Stavroulakis

Abstract

We obtain a quantitative form of Schur's comparison theorem for curves with finite total curvature in CAT(k) spaces. This sharpens and extends the classical arm and bow lemmas in Euclidean space, as well as their Riemannian analogues. The proof is based on a comparison formula for curves in model planes, expressed in terms of curvature measures and a notion of moment arm borrowed from mechanics. Another ingredient is a refinement of Reshetnyak's theorem that controls the curvature of the majorizing curve.

Keywords

Schur comparison theorem CAT(k) spaces curves total curvature curvature measures moment arm

Cite

@article{arxiv.2607.15106,
  title  = {A quantitative Schur comparison theorem for curves in CAT(k) spaces},
  author = {Mohammad Ghomi and John Ioannis Stavroulakis},
  journal= {arXiv preprint arXiv:2607.15106},
  year   = {2026}
}

Comments

26 pages, 5 figures