Effective Universal Coverings and Local Minima of the Length Functional on Loop Spaces

Abstract

In this paper we describe a construction of effective universal coverings of Riemannian manifolds (and, more generally, of path metric spaces). We give two applications of this construction.

To describe the first application let M ⁿ be a simply connected closed Riemannian manifold, and Ω _x M ⁿ denote the space of all loops based at a point ${x\in M^{n}}$. We demonstrate that if the length functional on Ω _x M ⁿ has a non-trivial very deep local minimum, then it has many deep local minima.

The second application is a quantitative version of a theorem proven by M. Anderson. To state this theorem assume that M ⁿ is a closed n-dimensional manifold with volume ≥ v > 0, diameter d and Ricci curvature ≥ −(n − 1). Then the theorem asserts that there exist explicit positive ${\epsilon = \epsilon(n, v, d)}$ and N = N(n, v, d) such that for each closed curve γ of length smaller than${\epsilon}$ one of its first N iterates is contractible. In the present paper we find an explicit upper bound for the length of curves in a homotopy contracting one of the first N iterates of γ.