Commutator formulas for gradient Ricci shrinker and their application to linear stability

TL;DR

This paper derives new commutator formulas for operators on gradient Ricci shrinkers and uses them to extend a theorem on the necessary conditions for their linear stability.

Contribution

It introduces new commutator formulas for operators on GRS+ metrics and generalizes a stability theorem by Cao and Zhu.

Findings

Derived commutator formulas for divergence and Laplacian operators on GRS+

Generalized a theorem on necessary conditions for linear stability of GRS+

Provides tools for analyzing stability of gradient Ricci shrinkers

Abstract

In this paper we have found some commutator formulas between $div_{f},\Delta_{f,L},div_{f}^\dagger,\Delta_{f}$ on closed orientable $GRS^+$ metrics (Gradient Ricci Shrinker), then with them we have generalized a Theorem of Cao and Zhu about necessary condition for linear stability of a $GRS^+$.