Complexity and variational problems in Rienmannian geometry

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The object of my study are Riemannian manifolds, which can be regarded as a generalization of surfaces in three-dimensionalspace for higher dimensions. I am interested in Riemannian manifolds that have an "optimal" shape, that is, are less curved than all nearby Riemannian manifolds. My other object of studyis geodesics on Riemannian manifolds. Geodesics are the straightest lines on Riemannian manifolds. (For example, geodesics on a plane are straight lines, geodesics on theround sphere are big circles, and so on). It is known that if a manifold is closed (like a sphere or a torus), then every two points can be connected by infinitely many distinct geodesics. These geodesics become longer and longer. I beleive and will try to prove that for every n one canestimate the length of the nth of these geodesics in terms of n and the maximal distance between points on the manifold. But no information about curvature of the manifold is necessary here. Finally, I am going to study shapes of closed geodesic nets. Geodesic nets are quite a mysterious class of objects that naturally appears, when one tries to figure out how to connect several phone stations (at different points of a surface) by the minimal amount of  cable.