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In this paper, we consider generalized Yamabe solitons which include many notions, such as Yamabe solitons, almost Yamabe solitons, $h$-almost Yamabe solitons, gradient $k$-Yamabe solitons and conformal gradient solitons. We completely classify the generalized Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the position vector field.

Carlotto, Chodosh and Rubinstein studied the rate of convergence of the Yamabe flow on a closed (compact without boundary) manifold $M$:

Studying the geometric flow plays a powerful role in mathematics and physics. We introduce the Yamabe flow on Finsler manifolds and we will prove the existence and uniqueness for solution of Yamabe flow on Berwald manifolds.

In this paper, we study conformal deformations and $C$-conformal deformations of Ricci-directional and second type scalar curvatures on Finsler manifolds. Then we introduce the best equation to study the Yamabe problem on Finsler manifolds. Finally, we restrict conformal deformations of metrics to $C$-conformal deformations and derive the Yamabe functional and the Yamabe flow in Finsler geometry.

In this paper, we first introduce the notion of almost quasi-Yamabe solitons and get some interesting formulas for them. Then, we explore conditions under which an almost quasi-Yamabe soliton is trivial and give some characterization results for it. Finally, we give a necessary and sufficient condition under which an arbitrary compact almost Yamabe soliton is necessarily gradiant.

In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several sufficient conditions on the soliton vector fields of Yamabe solitons under which their metrics are Yamabe metrics.