Revisiting gradient conformal solitons (Q6583349)

Let \((M, g)\) be an \(n\)-dimensional Riemannian manifold and let \(f\) and \(\phi\) be smooth functions defined on \(M\). The quadruple \((M, g, f, \phi)\) is called a gradient conformal soliton if it satisfies\N\[\N\nabla^{2} f = \phi g ,\N\]\Nwhere \(\nabla^{2}f\) stands for the Hessian of \(f\) with respect to the metric \(g\). In this case, the function \(f\) is called the potential function. When \(f\) is constant, \((M , g, f, \phi)\) is called trivial. \N\NUnder mild hypotheses on the potential function, the authors obtain new triviality results, for example:\N\begin{itemize}\N\item if \((M , g, f, \phi)\) is a complete noncompact \(n\)-dimensional gradient conformal soliton such that \(\mathrm{Ric}_g (\nabla f, \nabla f) \leq 0\) and \(|\nabla f|\) converges to zero at infinity; or\N\item if \((M , g, f, \phi)\) is a complete \(n\)-dimensional gradient conformal soliton with polynomial volume growth, \(\mathrm{Ric}_g (\nabla f, \nabla f) \leq-a|\nabla f|^2\), for some positive constant \(a\), and \(|\nabla f|, |\nabla^2 f| \in L^{\infty}(M)\); or\N\item if \((M , g, f, \phi)\) is a stochastically complete \(n\)-dimensional gradient conformal soliton such that \(\mathrm{Ric}_g (\nabla f, \nabla f)\leq-a|\nabla f|^2\), for some positive constant \(a\), and \(|\nabla f| \in L^{\infty}(M)\);\N\end{itemize}\Nthen in each case the gradient conformal soliton is trivial.