# American Institute of Mathematical Sciences

December  2019, 39(12): 7057-7080. doi: 10.3934/dcds.2019295

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

 1 School of Mathematical Sciences, Beijing Normal University, No. 19, Xinjiekouwai St., Haidian District, Beijing 100875, China 2 School of Mathematics, Georgia Institute of Technology, 686 Cherry St. Atlanta GA 30332, USA

To Luis Caffarelli in his 70th birthday, with admiration.

Received  October 2018 Revised  June 2019 Published  September 2019

Fund Project: X. S. supported by the National Natural Science Foundation of China (Grant No. 11871242).
R. L. has been partially supported by NSF grant DMS-1800241. Progress was made while R.L. was visiting the JLU-GT institute for theoretical Science and Beijing Normal University. The final version was written while R.L. was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester supported by DMS-1440140.

We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in [30,Section 5.11] 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, the 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.

Citation: Xifeng Su, Rafael de la Llave. On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7057-7080. doi: 10.3934/dcds.2019295
##### References:

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To Luis Caffarelli in his 70th birthday, with admiration.

##### References:
Heuristic picture of the Almgren-Federer metric for a fixed x
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