Sharp uncertainty principles on metric measure spaces

We prove the rigidity of the Heisenberg–Pauli–Weyl uncertainty principle and the Caffarelli–Kohn–Nirenberg interpolation inequality, on metric measure spaces satisfying measure contraction property. Non-trivial examples fitting our setting include Finsler manifolds with non-negative Ricci curvature and many ideal sub-Riemannian manifolds, such as Heisenberg groups, the Grushin plane and Sasakian structures.