Nabutovsky, A.; Rotman, R. Volume, diameter and the minimal mass of a stationary 1-cycle. (English) Zbl 1073.53057 Geom. Funct. Anal. 14, No. 4, 748-790 (2004). A well-known problem in differential geometry concerns upper bounds of the minimal length of closed geodesics of a compact manifold in terms of the volume of the manifold. A naturally related question is to bound it in terms of the diameter. In this paper, it is proven that the minimal length of stationary 1-cycles can be bounded in terms of the diameter. Closed geodesics are stationary 1-cycles, and hence this result is a weaker version of the bound on the minimal length of geodesics in terms of the diameter. Similar bounds are also obtained in terms of the filling radius, a notion introduced by Gromov. Reviewer: Lizhen Ji (Ann Arbor) Cited in 16 Documents MSC: 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C20 Global Riemannian geometry, including pinching Keywords:volume; diameter; stationary 1-cycle; filling radius PDFBibTeX XMLCite \textit{A. Nabutovsky} and \textit{R. Rotman}, Geom. Funct. Anal. 14, No. 4, 748--790 (2004; Zbl 1073.53057) Full Text: DOI arXiv