Quantum Limits on product manifolds

TL;DR

This paper investigates the behavior of quantum limits on product manifolds, showing conditions under which these limits are absolutely continuous and excluding certain geodesics from being charged.

Contribution

It establishes new properties of quantum limits on product manifolds, including absolute continuity transfer and restrictions on periodic geodesics.

Findings

Quantum limits on product manifolds are absolutely continuous if limits on each factor are.

Periodic geodesics cannot be charged by quantum limits on manifolds with the minimal multiplicity property.

Results extend understanding of quantum limits in geometric analysis.

Abstract

We establish some properties of quantum limits on a product manifold, proving for instance that, under appropriate assumptions, the quantum limits on the product of manifolds are absolutely continuous if the quantum limits on each manifolds are absolutely continuous. On a product of Riemannian manifolds satisfying the minimal multiplicity property, we prove that a periodic geodesic can never be charged by a quantum limit.