Compactness theorems for gradient Ricci solitons

In this paper, we prove the compactness theorem for gradient Ricci solitons. Let (Mα, gα) be a sequence of compact gradient Ricci solitons of dimension n ≥ 4, whose curvatures have uniformly bounded L n 2 norms, whose Ricci curvatures are uniformly bounded from below with uniformly lower bounded volume and with uniformly upper bounded diameter, then there must exists a subsequence (Mα, gα) converging to a compact orbifold (M∞, g∞) with finitly many isolated singularities, where g∞ is a gradient Ricci soliton metric in an orbifold sense. MSC: 58G30, 53C20