Rotationally symmetric gradient Yamabe solitons (Q6622532)

A Riemannian manifold \((M, g)\) is said to be a gradient Yamabe soliton if there exists a smooth function \(f: M\rightarrow \mathbb R\) and a scalar \(\lambda\) such that \(\nabla \nabla f=(R-\lambda)g,\) where \(\nabla \nabla f\) denotes the Hessian of \(f\) and \(R\) denotes the scalar curvature of \((M, g).\) \N\NThe authors investigate conditions under which a gradient Yamabe soliton has constant scalar curvature. After giving a new proof of the known fact that any compact gradient Yamabe soliton must have constant scalar curvature, the authors obtain the same conclusion in Theorem \(2\) for a complete, noncompact gradient Yamabe soliton \((M, g, f)\) under an additional integral condition on the potential function \(f.\) The authors also give a classification for \(3\)-dimensional gradient Yamabe solitons. Finally, similar questions for \(k\)-Yamabe solitons are considered.