Remarks on potential functions of noncompact quasi-Einstein manifolds

TL;DR

This paper investigates potential functions on noncompact quasi-Einstein manifolds, establishing bounds on their space and characterizing the manifold structure when maximum dimension is achieved, also relating asymptotic flatness to Ricci-flatness.

Contribution

It provides a classification of positive potential functions on three-dimensional noncompact quasi-Einstein manifolds and links asymptotic flatness with Ricci-flatness.

Findings

The space of positive potential functions has dimension at most two.

Equality in dimension implies the manifold is a product with a $ ext{λ}$-Einstein surface.

Asymptotically flat $ ext{λ}=0$ quasi-Einstein manifolds are Ricci-flat.

Abstract

In this article, we study the set of potential functions on noncompact quasi-Einstein manifolds. We show that the space of all positive potential functions on a three-dimensional noncompact quasi-Einstein manifold has dimension at most two, and that equality holds if and only if the manifold is isometric to a product $B\times\mathbb{R}$, where $B$ is a $\lambda$-Einstein surface or one of the examples obtained by L. Berard Bergery and described in Besse's book. Moreover, we prove that any asymptotically flat $n$-dimensional quasi-Einstein manifold with $\lambda=0$ is necessarily Ricci-flat.