Existence of finite global norm of potential vector field in a Ricci soliton

TL;DR

This paper studies conditions under which the potential vector field in Ricci solitons has finite global norm, revealing that in certain cases it implies constant scalar curvature.

Contribution

It establishes new conditions for the finiteness of the potential vector field's global norm in Ricci solitons and links this to scalar curvature constancy.

Findings

Finite global norm of potential vector field in expanding Ricci solitons under specific conditions.

Finite global norm implies constant scalar curvature in complete non-compact Ricci solitons with finite volume.

Abstract

In this article, we investigate global norm of potential vector field in Ricci soliton. In particular, we have deduced certain conditions so that the potential vector field has finite global norm in expanding Ricci soliton. We have also showed that if the potential vector field has finite global norm in complete non-compact Ricci soliton having finite volume, then the scalar curvature becomes constant.