AI 中文总结
研究通过嵌入圆的极小极大宽度探讨黎曼球面等几何问题，利用特定方法证明不变量下界，关联非紧流形嵌入圆宽度上界与函数存在性，界定欧氏平面曲线宽度，还证明了佐尔球面中圆球的刚性结果。
AI 中文摘要
我们证明了黎曼球面的伯克霍夫极小极大不变量关于其嵌入圆的极小极大宽度的一个下界。主要工具是一种方法，可从给定的由闭曲线对球面的扫掠诱导出嵌入圆中由点对构成的扫掠，使得诱导扫掠中每对点位于环境扫掠的相邻曲线中。此外，对于非紧完备黎曼流形，我们将其嵌入圆的极小极大宽度的上界与从该流形到有限图的连续函数的存在性相关联，这些函数的水平集具有一致有界直径，从而在乌里松定量维数理论中给出其1 -宽度的一个估计。在欧几里得平面的特定情形下，我们用其极小极大宽度从下方界定闭简单曲线的经典宽度。最后，我们证明了相关的刚性结果，刻画了佐尔球面中的圆球。
英文摘要
We prove a lower bound for the Birkhoff min-max invariant of a Riemannian sphere in terms of the min-max width of its embedded circles. The main tool is a method to induce a sweepout by pairs of points in an embedded circle from a given sweepout of the sphere by closed curves, so that the points of each pair of the induced sweepout lie close to two curves of the ambient sweepout that are close to each other. Moreover, considering a non-compact complete Riemannian manifold, we relate upper bounds for the min-max width of its embedded circles to the existence of continuous functions from the manifold to finite graphs with level sets that have uniformly bounded diameters, thus giving an estimate for its 1-width in Urysohn's quantitative dimension theory. In the specific case of the Euclidean plane, we bound from below the classical width of a simple closed curve by its min-max width. Finally, we prove related rigidity results characterizing the round spheres among Zoll spheres.
Comments32 pages, 1 figure. Minor revision, references added. Submitted version
