On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics

TL;DR

This paper explores Almgren-Federer's example in geometric measure theory, illustrating subtle issues in geodesic minimization and the distinctions between different notions of integrability and hyperbolicity in dynamical systems.

Contribution

It provides an exposition of Almgren-Federer's example from the perspective of geodesics, clarifying its implications for the theory of minimizing geodesics and dynamical systems.

Findings

Constructs metrics on  with non-Class-A Tonelli geodesics

Shows that some length-minimizing curves are not Class-A minimizers

Illustrates differences between various definitions of integrability and hyperbolicity

Abstract

We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in \cite[Section 5.11]{Federer74} to illustrate the need of certain definitions in the calculus of variations. The Almgren-Federer example, besides its intended goal of illustrating subtle aspects of geometric measure theory, is also a problem in the theory of geodesics. Hence, we wrote an exposition of the beautiful ideas of Almgren and Federer from the point of view of geodesics. In the language of geodesics, Almgren-Federer example constructs metrics in $\mathbb{S}^1\times \mathbb{S}^2$, with the property that none of the Tonelli geodesics (geodesics which minimize the length in a homotopy class) are Class-A minimizers in the sense of Morse (any finite length segment in the universal cover minimizes the length between the end points; this is also sometimes given other names). In other words, even if a curve is a minimizer of length among all the curves homotopic to it, by repeating it enough times, we get a closed curve which does not minimize in its homotopy class. In that respect, the example is more dramatic than a better known example due to Hedlund of a metric in $\mathbb{T}^3$ for which only 3 Tonelli minimizers (and their multiples) are Class-A minimizers. For dynamics, the example also illustrates different definitions of `integrable' and clarifies the relation between minimization and hyperbolicity and its interaction with topology.