| Tartalmi leírás: |
We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in $mathbb R^n$ (shortly, (it 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 (it non-positive sectional
curvature), the sharp HPW principle holds on $(M,g)$. However, (it positive extremals exist) in the sharp HPW
principle if and only if $(M,g)$ is
isometric to $mathbb R^n$, $n=(rm dim)(M)$.
(b) When $(M,g)$ has (it non-negative Ricci curvature), the sharp HPW principle holds on $(M,g)$ if and only if $(M,g)$ is isometric to $mathbb R^n$.
Since the sharp HPW principle and the Hardy-Poincar'e 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.
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