Integral pinched shrinking Ricci solitons (Q323669)

A Riemannian metric \(g\) on the manifold \(M\) is a gradient Ricci soliton if there exists a smooth function \(f\) such that NEWLINE\[NEWLINE\mathrm{Ric} + \nabla^2 f = \lambda gNEWLINE\]NEWLINE for some \(\lambda\in {\mathbb{R}}\). Depending on whether \(\lambda>0\), \(\lambda =0\) or \(\lambda<0\), the soliton is shrinking, steady or expanding. Hence, Ricci solitons are natural generalizations of Einstein metrics and they are important in many situations in physics.NEWLINENEWLINEIn the present paper, compact Ricci solitons are considered. It was shown in previous works that there are no nontrivial compact steady or expanding Ricci solitons. Further, in dimension \(3\), compact shrinking Ricci solitons are quotients of the standard sphere. In dimension \(4\), the first nontrivial examples were found. Also cases of compact shrinking Ricci solitons satisfying pointwise curvature conditions were investigated (in general dimension) and many of them are quotients of the sphere.NEWLINENEWLINEIn the present paper, compact shrinking Ricci solitons satisfying integral curvature conditions are investigated and it is shown that they are quotients of the sphere. Special attention is paid to dimension \(4\), however some results are applicable also in general dimension.