Uncertainty principle and its rigidity on complete gradient shrinking Ricci solitons

Abstract

We prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in R n \mathbb {R}^n . In addition, we partially give analogous rigidity results of the Caffarelli-Kohn-Nirenberg inequalities on shrinking Ricci solitons.