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Sharp uncertainty principles on general Finsler manifolds. (English) Zbl 1452.53063

Summary: The paper is devoted to sharp uncertainty principles (Heisenberg-Pauli-Weyl, Caffarelli-Kohn-Nirenberg, and Hardy inequalities) on forward complete Finsler manifolds endowed with an arbitrary measure. Under mild assumptions, the existence of extremals corresponding to the sharp constants in the Heisenberg-Pauli-Weyl and Caffarelli-Kohn-Nirenberg inequalities fully characterizes the nature of the Finsler manifold in terms of three non-Riemannian quantities, namely, its reversibility and the vanishing of the flag curvature and \(S\)-curvature induced by the measure, respectively. It turns out in particular that the Busemann-Hausdorff measure is the optimal one in the study of sharp uncertainty principles on Finsler manifolds. The optimality of our results are supported by Randers-type Finslerian examples originating from the Zermelo navigation problem.

MSC:

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
26D10 Inequalities involving derivatives and differential and integral operators
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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