On Nilpotent and Solvable Quasi-Einstein Manifolds

In this paper, we investigate nilpotent and unimodular solvable Lie groups that admit quasi-Einstein metrics $(M,g,X)$ with $X$ a left-invariant vector field, which we call \emph{totally left-invariant quasi-Einstein metrics}.

We give a complete classification of nilpotent Lie groups admitting such metrics, proving that this occurs if and only if the group is isomorphic to a Heisenberg Lie group.

For unimodular solvable Lie groups $S$, we show that the existence of a non-flat totally left-invariant quasi-Einstein metric forces the center of $S$ to be one-dimensional.

Furthermore, under the additional assumption that the adjoint action $\operatorname{ad}_a$ of $S$ is a normal derivation, we obtain a full classification: these groups are standard and their nilradical must be a Heisenberg Lie algebra.

As an application, we prove that the only near-horizon geometries on a compact nilmanifold are $\Gamma \backslash H_{n}$, where $ H_{n}$ is $n$-dimensional Heisenberg Lie group.