A quantitative Schur comparison theorem for curves in CAT(k) spaces
arXiv:2607.15106
summary
The paper provides a quantitative version of Schur's comparison theorem for curves with finite total curvature in CAT(k) spaces, extending classic results from Euclidean and Riemannian geometry.
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.
26 pages, 5 figures
Topics & keywords
#cat(k) spaces#schur comparison theorem#curve curvature#geometric analysis#comparison theoremstotal curvaturemodel planesmoment armReshetnka ymajorizing curve