Isometry theorem of gradient Shrinking Ricci solitons

Abstract: In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic, then the manifold is isometric to the Euclidean sphere. As a consequence, we have showed that a four dimensional gradient shrinking Ricci soliton satisfying some conditions is isometric to $\mathbb{S}^4$ or $\mathbb{RP}^4$ or $\mathbb{CP}^2$. We have also deduced a condition for the shrinking Ricci soliton to be compact with quadratic volume growth.