Introduction:
Let $(M^n,g)$ be Riemannian manifold and consider a smooth function $f:M\to\left(0,+\infty\right)$.
The $m$-Bakry–Emery–Ricci tensor is defined as
$$
Ric_f^m:=Ric+\nabla^2f-\frac{1}{m}df\otimes df,\quad 0< m\leq \infty,
$$
where $Ric$ is the $(0,2)$ Ricci tensor of $(M,g)$ and $\nabla^2$ is the Hessian.
A triple $(M,g,f)$ is called $m$-quasi Einstein if there exists $\lambda\in\mathbb{R}$ such that
\begin{equation}
Ric_f^m=\lambda g\label{eq: quasi E}.
\end{equation}
When $0<m<\infty$ one can consider $u=e^{-f/m}$ and using the relations ($\nabla u$ is the gradient of $u$)
\begin{align*}
\nabla u&=-\frac{1}{m}e^{-f/m}\nabla f\\
\frac{m}{u}\nabla^2 u&=-\nabla^2 f+\frac{1}{m} df\otimes df,
\end{align*}
one sees
\begin{equation}
Ric_f^m=\lambda g\Longleftrightarrow Ric-\frac{m}{u}\nabla^2 u=\lambda g.
\end{equation}
Taking the trace of the last relation one obtains
\begin{equation}
\Delta u=\frac{u}{m}(S-\lambda n)
\end{equation}
where $S$ is the scalar curvature of $(M,g)$
I am trying to understand:
In "Rigidity of quasi-Einstein metrics", the authors claim that since $u>0$, the last above equation immediately implies (Proposition 2.1):
A compact quasi-Einstein metric with constant scalar curvature is
trivial, i.e, $f$ is constant.
(1) I would like to know a justification for that claim.
I only realized that under the condition that the scalar curvature $S$ is constant then $u=e^{-f/m}$ must be an eigenfunction of the Laplacian with eigenvalue $\frac{S-\lambda n}{m}$ then I cannot conclude anymore. I don't know if a "known" result is needed.
(2) I would like to list/collect other sufficient conditions to conclude that a $m$-quasi Einstein metric is trivial.
Best Answer
(1) The key is that $u$ is positive (as the exponential of something). As you already noted, $u$ is an eigenfunction of the Laplacian. But the strong maximum principle implies that (on a compact manifold) the only positive eigenfunctions of the Laplacian are the positive constants.
(2) The literature now contains far more sufficient conditions for the triviality of an $m$-quasi Einstein metric than I can keep track of. If you have access to MathSciNet, you can find all published papers which cite our paper, many of which contain some new sufficient condition. You can also use Google scholar for a similar purpose. As far as I know, only in dimensions one or two do we know exactly which Riemannian manifolds are $m$-quasi-Einstein; see this article of He, Petersen and Wylie.