# Uncertainty Principle and its rigidity on complete gradient shrinking Ricci solitons

## TL;DR

This paper establishes rigidity theorems for shrinking gradient Ricci solitons supporting the uncertainty principle and related inequalities, revealing their geometric structure and constraints.

## Contribution

It proves new rigidity results for shrinking gradient Ricci solitons supporting the uncertainty principle with sharp constants and extends analogous results to Caffarelli-Kohn-Nirenberg inequalities.

## Key findings

- Rigidity theorems for Ricci solitons supporting the uncertainty principle.
- Partial results on Caffarelli-Kohn-Nirenberg inequalities on Ricci solitons.
- Characterization of geometric structures under these inequalities.

## Abstract

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