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Let $\mathcal{C}(\mathcal{R},n,p,\Lambda,D,V_0)$ be the class of compact$n$-dimensional Riemannian manifolds with finite diameter $\leq D$,non-collapsing volume $\geq V_0$ and $L^p$-bounded $\mathcal{R}$-curvaturecondition $\|\mathcal{R}\|_{L^p}\leq \Lambda$ for some $p>\frac n2$. Let$(M,g_0)$ be a compact Riemannian manifold and $\mathcal{C}(M,g_0)$ the classof manifolds $(M,g)$ conformal to $(M,g_0)$. In this paper we use$\varepsilon$-regularity to show a rigidity result in the conformal class$\mathcal{C}(S^n,g_0)$ of standard sphere under $L^p$-scalar rigiditycondition. Then we use harmonic coordinate to show $C^{\alpha}$-compactness ofthe class $\mathcal{C}(K,n,p,\Lambda,D,V_0)$ with additional positive Yamabeconstant condition, where $K$ is the sectional curvature, and this result willimply a generalization of Mumford's lemma. Combining these methods together wegive a geometric proof of $C^{\alpha}$-compactness of the class$\mathcal{C}(K,n,p,\Lambda,D,V_0)\cap \mathcal{C}(M,g_0)$. By using Weyl tensorand a blow down argument, we can replace the sectional curvature condition byRicci curvature and get our main result that the class$\mathcal{C}(Ric,n,p,\Lambda,D,V_0)\cap \mathcal{C}(M,g_0)$ has$C^{\alpha}$-compactness.

Convergence of manifolds under some $L^p$-integral curvature conditions