Unexpected Analytic Phenomena on Finsler Manifolds

TL;DR

This paper reveals unexpected phenomena on Finsler manifolds, showing that classical inequalities and embedding theorems fail or change behavior dramatically compared to the Riemannian case, especially on flat Finsler spaces.

Contribution

It demonstrates that key geometric and functional analysis properties differ fundamentally on Finsler manifolds, contrasting with classical Riemannian results, and identifies the role of $S$-curvature in these differences.

Findings

Nash embedding theorem fails on flat Finsler spaces.

Sobolev spaces become nonlinear on certain Finsler manifolds.

Hardy and uncertainty inequalities break down, but CKN inequality has a sharp threshold.

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.