The main goal of this proposal is to establish curvature-free estimates for the length/area/volume of minimal objects in Riemannian geometry, such as (not the shortest) geodesic segments, closed geodesics, geodesic loops, minimal surfaces, etc. In particular, J. P. Serre had shown that for any two points on a closed Riemannian manifold there exist infinitely many geodesic segments joining them. While it is a trivial statement that the length of a shortest geodesic segment joining any two points equals to at most the diameter of a manifold, the investigator proposes to study the lengths of the other geodesic segments. For example, it would be interesting to find out whether there always exist n distinct geodesic segments of length at most n times the diameter of a closed manifold.
Riemannian geometry is a generalization of geometry of surfaces to higher dimensions. Some of the central objects of study of Riemannian geometry are geodesics, geodesic nets, minimal surfaces and other minimal objects. Geodesics generalize the notion of a straight line to Riemannian geometry. The two essential properties of a line are: (1) that it is straight; (2) that it minimizes distance between two points. We define geodesic as "the straightest" curve that lies on a given Riemannian manifold. It turns out that the distance between two points is minimized by a geodesic segment. However, the analogy between geodesic and line segments is not perfect. For example, in the case of a closed Riemannian manifold there are infinitely many geodesic segments connecting any two points, as it was shown by J. P. Serre. Also, some geodesics close on themselves and become periodic. The investigator proposes to study the connection between the lengths of various geodesic segments connecting any two points on a closed Riemannian manifold and the size of this manifold represented by its volume and/or the diameter, defined as the maximal distance between two points. The investigator also proposes to study the relationship between the length of a shortest closed geodesic on a manifold and the size of a manifold, as well as relations between the size of a manifold and various other minimal objects.
