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 R n (shortly, sharp HPW principle). Our results deeply depend on the curvature of the Riemannian manifold which can be roughly formulated as follows:

An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds

“…Though, except for N = 3, the optimality of the weight V does not imply that N − 2 is the best constant in obvious sense. It is also interesting to note that our optimal inequality is closely related to the improved Hardy inequality studied in [31], we refer to Remark 2.2 for a detailed discussion. Here we only mention that the main result on the Hardy inequality given in [31], when considered on H N , follows as a particular case of our results.…”

“…It is also interesting to note that our optimal inequality is closely related to the improved Hardy inequality studied in [31], we refer to Remark 2.2 for a detailed discussion. Here we only mention that the main result on the Hardy inequality given in [31], when considered on H N , follows as a particular case of our results. Also the extension of our results to more general Cartan-Hadamard manifolds is obtained under less restrictive assumptions than in [31].…”

“…Much of the interest has centered on optimal improvements of the inequality and the effect of the domain on the Hardy constant. Its generalization to Riemannian manifolds was intensively pursued after the seminal work of Carron [15], see for instance [5,7,18,29,30,31,40]. Let (M, g) be a Riemannian manifold and let ̺(x) be a weight function satisfying the Eikonal equation |∇ g ̺| = 1 and ∆ g ̺ ≥ C ̺ where C > 0 a positive constant.…”

“…Existence/nonexistence of extremals for the related Hardy-Poincaré inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator P λ .In case of Cartan-Hadamard manifold M of dimension N (namely, a manifold which is complete, simply-connected, and has everywhere non-positive sectional curvature), the geodesic distance function d(x, x 0 ), where x 0 ∈ M , satisfies all the assumptions of the weight ̺ and the above inequality holds with best constant N −2 2 2 , see [31]. In particular, consid-Date: November 23, 2017.…”

Interpolation between Brezis–Vázquez and Poincaré inequalities on nonnegatively curved spaces: sharpness and rigidities

“…More precisely, such Sobolev-type inequalities behave quite naturally on Hadamard manifolds (simply connected, complete Riemannian/Finsler manifolds with nonpositive sectional/flag curvature), as shown e.g. by Carron [7,8], Berchio, Ganguly and Grillo [5], D'Ambrosio and Dipierro [10], Kombe andÖzaydin [16,17], Farkas, Kristály and Varga [13], Kristály [18], Yang, Su and Kong [26]. This fact is not surprising since Hadamard manifolds are diffeomorphic to Euclidean spaces.…”