Convergence Of Manifolds Under Some L~p-integral Curvature Conditions

Let C（R,n,p,Λ,D,V₀）be the class of compact n-dimensional Riemannian manifolds with finite diameter≤D,non-collapsing volume≥V₀and L^p-bounded R-curvature and C（M,g₀）the class of manifolds（M,g）conformal to（M,g₀）.In this paper we useε-regularity to show a rigidity result in the conformal class C（Sⁿ,g₀）of standard sphere under L^p-scalar rigidity condition.Then we use harmonic coordinate to show C^α-compactness of the class C（K,n,p,Λ,D,V₀）with additional positive Yamabe constant condition,where K is the sectional curvature,and this result will imply a generalization of Mumford’s lemma.Combining these methods together we give a geometric proof of C^α-compactness of the class C（K,n,p,Λ,D,V₀）∩C（M,g₀）.By using Weyl tensor and a blow down argument,we can replace the sectional curvature condition by Ricci curvature and get our main result that the class C（Ric,n,p,Λ,D,V₀）∩C（M,g₀）has C^α-compactness.