Integral pinching characterization of compact shrinking Ricci solitons

We investigate the pinching problem for shrinking compact Ricci solitons. Firstly, we show that every $n$-dimensional $(n\ge 4)$ shrinking compact Ricci soliton $(M^n,g)$ is isometric to a finite quotient of $\mathbb S^n$ 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 $p \gt 2/n$. The arguments rely mainly on algebraic curvature estimates and several important integral inequalities.