Sharp uncertainty principles on Riemannian manifolds: the influence of curvature

Abstract

We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg–Pauli–Weyl uncertainty principle with the sharp constant in Rn (shortly, sharp HPW principle). Our results deeply depend on the curvature of the Riemannian manifold which can be roughly formulated as follows:(a)When (M,g) has non-positive sectional curvature, the sharp HPW principle holds on (M,g). However, positive extremals exist in the sharp HPW principle if and only if (M,g) is isometric to Rn, n=dim(M).(b)When (M,g) has non-negative Ricci curvature, the sharp HPW principle holds on (M,g) if and only if (M,g) is isometric to Rn.Since the sharp HPW principle and the Hardy–Poincaré inequality are endpoints of the Caffarelli–Kohn–Nirenberg interpolation inequality, we establish further quantitative results for the latter inequalities in terms of the curvature on Cartan–Hadamard manifolds.