Example of quasi Yamabe gradient soliton

A $(M,g)$ Riemannian manifold is called quasi Yamabe gradient soliton if there exists a smooth function $f\in C^\infty(M)$ such that the following condition holds $$Hess(f)=(R-\lambda)g+\mu df\otimes df,$$ where $R$ is the scalar curvature of $g$ and $\lambda,\mu$ are constants. The concept of quasi Yamabe soliton was first introduced by Huang, Guangyue; Li, Haizhong, On a classification of the quasi Yamabe gradient solitons, Methods Appl. Anal. 21, No. 3, 379-390 (2014). ZBL1304.53033.
But I am not able to find any nontrivial example of quasi Yamabe gradient soliton in Euclidean manifold with some proper metric. In the paper, Wang, Lin Feng, On noncompact quasi Yamabe gradient solitons, Differ. Geom. Appl. 31, No. 3, 337-348 (2013). ZBL1279.53039, there is an example in warped product manifold but I need in Euclidean space.

Please help me to find an example of that. Thank you

Solution 1:

See the preprint https://arxiv.org/abs/2106.10833. In this paper there are examples of compact and noncompact quasi k-Yamabe solitons.

Abstract. In this paper, we show that any compact quasi $k$-Yamabe gradient solitons must have constant $σ_k$-curvature. Moreover, we provide a certain condition for a compact quasi $k$-Yamabe soliton to be gradient, and for noncompact solitons, we present a geometric rigidity under a [decay] condition on the norm of the soliton field.