2025/06/16 by Jakub Takáč, Takáč, Jakub
Engineering · Mathematics · #35B65 (Secondary) #49Q15 (Primary) 26B10 #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #Control and Dynamics of Mobile Robots #Dynamics and Control of Mechanical Systems #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2506.13718
openalex publication_date 2025/06/16 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/31
We show that Lang's Flat Chain Conjecture (that is, without requiring finite mass of the underlying currents) fails for metric k-currents in ℝd whenever d≥ 2 and k∈\1, …, d\. In all other cases, it holds. The original conjecture due to Ambrosio and Kirchheim remains open. We first connect Lang's conjecture to a regularity statement concerning the prescribed Jacobian equation near L^∞. We then show that the equation does not have the required regularity. For a Lipschitz vector field π, its derivative Dπ exists a.e. and is identified with a matrix. Our non-regularity results for the prescribed Jacobian equation quantify how "small" the set conv(\detD π: Lip(π)≤ L\)⊂ L^∞ is for every L>0. The symbol "conv" stands for the convex hull. The "smallness" is quantified in topological terms and is used to show that Lang's Flat Chain Conjecture fails.