Integral pinching characterization of compact shrinking Ricci solitons

Abstract

We investigate the pinching problem for shrinking compact Ricci solitons. Firstly, we show that every n-dimensional (n ≥ 4) shrinking compact Ricci soliton (M n , g) is isometric to a finite quotient of S n under an L n/2 -pinching condition. Then we prove that the same result is still true for (M n , g) under an L p -pinching condition for p > 2/n. The arguments rely mainly on algebraic curvature estimates and several important integral inequalities.