An uncertainty principle on compact manifolds

TL;DR

This paper introduces a broad family of uncertainty principles on compact manifolds, unifying classical results and posing new geometric questions about optimality and embeddings.

Contribution

It generalizes existing uncertainty principles to arbitrary compact manifolds, linking them to geometric embedding problems and opening new research directions.

Findings

Includes classical uncertainty principles as special cases

Proposes a new geometric problem related to embeddings and uncertainty

Discusses open problems and extensions to disconnected 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 give 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.