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Volume, diameter and the minimal mass of a stationary 1-cycle. (English) Zbl 1073.53057

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.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C20 Global Riemannian geometry, including pinching
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