Unexpected Analytic Phenomena on Finsler Manifolds
Benling Li, Wei Zhao
TL;DR
The paper shows that flat Cartan–Hadamard Finsler manifolds exhibit radically different functional-analytic behavior from the Riemannian case: Berwald's flat metric destroys linear Sobolev structure and defeats Hardy, uncertainty, and HPW-type results, while preserving a sharp s-threshold for a generalized CKN inequality. In contrast, Funk metric spaces maintain many non-Riemannian features but lack this threshold, though Hardy and uncertainty-type inequalities still fail; these divergences are traced to the distinct behavior of the S-curvature. A general curvature-dominated framework is developed (via S-curvature and Ricci bounds) that explains when these inequalities fail on Finsler manifolds and highlights the crucial role of S-curvature in determining the validity of Sobolev-type inequalities on non-Riemannian spaces. The results underscore the delicate dependence of functional-analytic tools for PDEs on the underlying Finsler geometry and point to the need for new methods in this broader setting.
Abstract
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.