Triviality Results and Conjugate Radius Estimation of Ricci Solitons

TL;DR

This paper studies Ricci solitons, proving triviality and rigidity results for compact cases, estimating conjugate radii for non-compact cases, and establishing properties related to curvature and potential fields.

Contribution

It provides new triviality and rigidity theorems for Ricci solitons, along with bounds on conjugate radius and properties of non-shrinking solitons with specific potentials.

Findings

Proved triviality for certain compact gradient Ricci solitons.

Derived a rigidity result for compact gradient shrinking Ricci solitons.

Estimated conjugate radius for non-compact gradient shrinking Ricci solitons.

Abstract

The investigation of Ricci solitons is the focus of this work. We have proved triviality results for compact gradient Ricci soliton under certain restriction. Later, a rigidity result is derived for a compact gradient shrinking Ricci soliton. Also, we have estimated the conjugate radius for non-compact gradient shrinking Ricci solitons with superharmonic potential. Moreover, an upper bound for the conjugate radius of Ricci soliton with concircular potential vector field is determined. Finally, it is proved that a non-compact gradient Ricci soliton with a pole and non-negative Ricci curvature is non-shrinking.