CAT(k)空间中曲线的定量舒尔比较定理
A quantitative Schur comparison theorem for curves in CAT(k) spaces
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中文总结 AI 辅助
研究CAT(k)空间中有限总曲率曲线的舒尔比较定理，基于模型平面曲线比较公式及相关概念，结合雷谢特尼亚克定理改进，得到定量形式，改进并扩展了经典引理。
中文摘要 AI 辅助
我们在CAT(k)空间中得到了具有有限总曲率曲线的舒尔比较定理的定量形式。这改进并扩展了欧几里得空间中的经典臂和弓引理及其黎曼类似物。证明基于模型平面中曲线的比较公式，该公式用曲率测度和从力学借用的力臂概念表示。另一个要素是对控制主曲线曲率的雷谢特尼亚克定理的改进。
英文摘要
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.
