Failure of famous functional inequalities on Finsler manifolds: the influence of S-curvature
摘要
The validity of functional inequalities on Finsler metric measure manifolds is based on three non-Riemannian quantities, namely, the reversibility, flag curvature and S-curvature induced by the measure. Under mild assumptions on the reversibility and flag curvature, it turned out that famous functional inequalities—as Hardy inequality, Heisenberg–Pauli–Weyl uncertainty principle and Caffarelli–Kohn–Nirenberg inequality—usually hold on forward complete Finsler manifolds with non-positive S-curvature. In this paper however we prove that—under similar assumptions on the reversibility and flag curvature as before—the aforementioned functional inequalities fail whenever the S-curvature is positive. Accordingly, our results clearly reveal the deep dependence of functional inequalities on the S-curvature. As a consequence of these results, we establish analytic aspects of Finsler manifolds, e.g., if the flag curvature is non-positive, the Ricci curvature is bounded from below and the S-curvature is positive, then the reversibility turns out to be infinite. Further topological properties and examples are presented on general Funk metric spaces, where the S-curvature plays again a decisive role.