Compactness theorems for gradient Ricci solitons

In this paper, we prove a compactness theorem for gradient Ricci solitons. Let (Mα,gα)(Mα,gα) be a sequence of compact gradient Ricci solitons of dimension n≥4n≥4, whose curvatures have uniformly bounded Ln2 norms, whose Ricci curvatures are uniformly bounded from below, with uniformly lower bounded volume and with uniformly upper bounded diameter; then there must exist a subsequence (Mα,gα)(Mα,gα) converging to a compact orbifold (M∞,g∞)(M∞,g∞) with finitely many isolated singularities, where g∞g∞ is a gradient Ricci soliton metric in an orbifold sense.