Ricci solitons from the perspectives of energy function

Abstract

This article explores to what extent the geometry of gradient Ricci solitons extends to non-gradient Ricci solitons. The primary tool is the energy function $E$ of the soliton. We study consequences of various bounds on $E$. Under mild assumptions on the scalar curvature, we prove a weighted $L^1$-Liouville type theorem for both the usual Laplacian and the drifted Laplacian $\Delta_V$ associated to soliton vector field $V$, the former of which implies that Ricci solitons with bounded energy function have at most one nonparabolic end. Finally, we show that the measure $e^{-E} \mathrm{d} \mathrm{Vol}_{g}$ is finite for complete shrinking Ricci solitons, partially generalizing a result of Aaron Naber. As a consequence, non-gradient shrinking Ricci solitons also have finite fundamental groups, as in the gradient case.