Complexity and Variational Problems in Differential Geometry

I am interested in quantitative aspects of Geometric Calculus

of Variations. In 1951 J.P. Serre proved that every two points

on a closed Riemannian manifold can be connected by an infinite set

of distinct geodesics. I would like to prove that the lengths of the first

k of them admit an upper bound of the form f(n,k)d, where n is the dimension

and d is the diameter of the manifold. I am interested in similar

curvature-free upper bounds for the length of the shortest periodic geodesic

and the smallest area of a minimal surface. I am also interested

in distribution of geodesic segments between a fixed pair of points

and geodesic nets on a manifold. In another direction I would like to extend

my previous results on fractal features of Morse

landscapes of Riemannian functionals to scale-invariant Riemannian

functionals involving a lower bound for the Ricci curvature. This would

involve proving some new results about Riemannian manifolds with Ricci

curvature bounded from below.

The notion of a closed Riemannian manifold is a higher dimensional

generalization of a closed surface, like the surface of a donut, or a sphere.

We plan to study connections between ``sizes' of various extremal

objects 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), geodesic

nets (objects that arize when one tries to connect three or more points

by a shortest tree) and minimal surfaces (i.e. mathematical models

of soap bubbles). In another direction we plan to study

'optimal' shapes of higher dimensional manifolds. Our approach to this last

question involves ideas coming from different areas of Mathematics,

including Computability Theory.