Ricci solitons from the perspectives of energy function

Abstract

In this article, we explore to what extent the geometry of gradient Ricci solitons can be carried over to the geometry of non-gradient Ricci solitons. We use energy function $E$ of the soliton as a tool to study this. We use Omori-Yau maximum principle to investigate the bounds on the energy function. One of the main results obtained is that a non-steady Ricci soliton with symmetric covariant derivative is gradient. We explore non-gradient Ricci solitons of constant scalar curvature. We prove a weighted $L^1$-Liouville type theorem with respect to the energy function under mild assumptions on the scalar curvature. Finally, we show that measure $e^{-E} d \mathrm{Vol}_{g}$ for complete shrinking Ricci soliton is finite, partially generalizing a result by Aaron Naber. Subsequently, this implies that nongradient shrinking Ricci solitons also have finite fundamental groups similar to their gradient counterparts.