On gradient Ricci solitons (Q1948721)

The authors study gradient Ricci solitons, i.e., complete Riemannian metrics on a smooth manifold \(M\), such that \[ \mathrm{Ric}+\mathrm{Hess}(f)=\rho g \] for some \(f\in C^{\infty}(M)\) and \(\rho\in\mathbb R\). A gradient Ricci soliton is called shrinking, steady or expanding if \(\rho>0\), \(\rho=0\), \(\rho<0\) respectively. The authors prove that for any complete gradient Ricci soliton \((M,g)\) we have \(\int_M|\mathrm{Ric}|^2e^{-\lambda f}<\infty\) and that if for some \(\lambda<1\) \(\int_M|\mathrm{Rm}|^2e^{-\lambda f}<\infty\) then \(\int_M|\nabla\mathrm{Ric}|^2e^{-f}=\int_M|\mathrm{div}(\mathrm{Rm})|^2e^{-f}<\infty\). In particular they obtain a classification of complete gradient shrinking Ricci solitons with harmonic Weyl tensor. The authors also prove that if \((M,g)\) is a gradient shrinking Kähler-Ricci soliton and \(u\) is a harmonic function on \(M\) such that \(\int_M|\nabla u|^2<\infty\) then \(u\) is constant. Also, some useful facts about steady Ricci solitons are proved in the paper.