2018/10/24 by Xifeng Su, Su, Xifeng, Rafael de la Llave +1
Mathematics · Physics and Astronomy · #34C25 #37J35 #37J50 #49Q15 #53C22 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1810.10508
openalex publication_date 2018/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathbbS1× \mathbbS2, 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 \mathbbT3 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.