Complexity and Variational Problems in Differential Geometry

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I am interested in quantitative aspects of Geometric Calculusof Variations. In 1951 J.P. Serre proved that every two pointson a closed Riemannian manifold can be connected by an infinite setof distinct geodesics. I would like to prove that the lengths of the firstk of them admit an upper bound of the form f(n,k)d, where n is the dimensionand d is the diameter of the manifold. I am interested in similar curvature-free upper bounds for the length of the shortest periodic geodesicand the smallest area of a minimal surface. I am also interestedin distribution of geodesic segments between a fixed pair of pointsand geodesic nets on a manifold. In another direction I would like to extendmy previous results on fractal features of Morselandscapes of Riemannian functionals to scale-invariant Riemannianfunctionals involving a lower bound for the Ricci curvature. This wouldinvolve proving some new results about Riemannian manifolds with Riccicurvature bounded from below.The notion of a closed Riemannian manifold is a higher dimensionalgeneralization of a closed surface, like the surface of a donut, or a sphere.We plan to study connections between ``sizes" of various extremalobjects on a closed Riemannian manifold and the ``size" of the manifold.Examples of extremal objects include geodesic segments (i.e.straightest curves between two points), periodic geodesics (i.e.straight curves on manifolds that smoothly close on themselves), geodesicnets (objects that arize when one tries to connect three or more pointsby a shortest tree) and minimal surfaces (i.e. mathematical modelsof soap bubbles). In another direction we plan to study"optimal" shapes of higher dimensional manifolds. Our approach to this lastquestion involves ideas coming from different areas of Mathematics,including Computability Theory.