An Uncertainty Principle on Compact Manifolds

Abstract

Breitenberger’s uncertainty principle on the torus $\mathbb {T}$ and its higher-dimensional analogue on $\mathbb {S}^{d-1}$ are well understood. We describe an entire family of uncertainty principles on compact manifolds $(M,g)$, which includes the classical Heisenberg–Weyl uncertainty principle (for $M=B(0,1) \subset \mathbb {R}^d$ the unit ball with the flat metric) and the Goh–Goodman uncertainty principle (for $M=\mathbb {S}^{d-1}$ with the canonical metric) as special cases. This raises a new geometric problem related to small-curvature low-distortion embeddings: given a function $f:M \rightarrow \mathbb {R}$, which uncertainty principle in our family yields the best result? We give a (far from optimal) answer for the torus, discuss disconnected manifolds and state a variety of other open problems.