TAILIEUCHUNG - Đề tài " Manifolds with positive curvature operators are space forms "

The Ricci flow was introduced by Hamilton in 1982 [H1] in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact four-manifolds with positive curvature operators are spherical space forms as well [H2]. More generally, the same conclusion holds for compact four-manifolds with 2-positive curvature operators [Che]. Recall that a curvature operator is called 2-positive, if the sum of its two smallest eigenvalues is positive. . | Annals of Mathematics Manifolds with positive curvature operators are space forms By Christoph B ohm and Burkhard Wilking Annals of Mathematics 167 2008 1079-1097 Manifolds with positive curvature operators are space forms By Christoph Bohm and Burkhard Wilking The Ricci flow was introduced by Hamilton in 1982 H1 in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact four-manifolds with positive curvature operators are spherical space forms as well H2 . More generally the same conclusion holds for compact four-manifolds with 2-positive curvature operators Che . Recall that a curvature operator is called 2-positive if the sum of its two smallest eigenvalues is positive. In arbitrary dimensions Huisken Hu described an explicit open cone in the space of curvature operators such that the normalized Ricci flow evolves metrics whose curvature operators are contained in that cone into metrics of constant positive sectional curvature. Hamilton conjectured that in all dimensions compact Riemannian manifolds with positive curvature operators must be space forms. In this paper we confirm this conjecture. More generally we show the following Theorem 1. On a compact manifold the normalized Ricci flow evolves a Riemannian metric with 2-positive curvature operator to a limit metric with constant sectional curvature. The theorem is known in dimensions below five H3 H1 Che . Our proof works in dimensions above two we only use Hamilton s maximum principle and Klingenberg s injectivity radius estimate for quarter pinched manifolds. Since in dimensions above two a quarter pinched orbifold is covered by a manifold see Proposition our proof carries over to orbifolds. This is no longer true in dimension two. In the manifold case it is known that the normalized Ricci flow converges to a metric of constant curvature for any initial metric H3 Cho . However there .

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