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.
|Effective start/end date
|8/15/07 → 7/31/10
- National Science Foundation: $144,720.00