Unexpected Analytic Phenomena on Finsler Manifolds

In the Riemannian setting, every flat Cartan–Hadamard manifold is isometric to Euclidean space, the canonical model that underlies the theory of Sobolev spaces and guarantees the sharpness/rigidity of the Hardy inequality, the uncertainty principle, and the Caffarelli–Kohn–Nirenberg (CKN) inequality. In this paper, we show that on a flat Finsler Cartan–Hadamard manifold – Berwald’s metric space – the classical picture alters radically: the Nash embedding theorem fails, the Sobolev space becomes nonlinear, and the Hardy and uncertainty inequalities break down completely, whereas the CKN inequality exhibits a sharp threshold in its validity depending on a parameter. By contrast, on Funk metric spaces – another class of Finsler Cartan–Hadamard manifolds – this threshold behavior disappears, although all the other non-Riemannian features persist. We trace this divergence to the lower bound of the $S$-curvature. As a consequence, the failure of the aforementioned functional inequalities is established for a broad class of Finsler manifolds.