tailieunhanh - Đề tài " Prescribing symmetric functions of the eigenvalues of the Ricci tensor "

We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric. 1. Introduction Let (M n , g) be a smooth, closed Riemannian manifold of dimension n. | Annals of Mathematics Prescribing symmetric functions of the eigenvalues of the Ricci tensor By Matthew J. Gursky and Jeff A. Viaclovsky Annals of Mathematics 166 2007 475 531 Prescribing symmetric functions of the eigenvalues of the Ricci tensor By Matthew J. Gursky and Jeff a. Viaclovsky Abstract We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature provided the conformal class admits an admissible metric. 1. Introduction Let Mn g be a smooth closed Riemannian manifold of dimension n. We denote the Riemannian curvature tensor by Riem the Ricci tensor by Ric and the scalar curvature by R. In addition the Weyl-Schouten tensor is defined by A - rìc 1 RgẦ. 1 n 2 V 2 n 1 w This tensor arises as the remainder in the standard decomposition of the curvature tensor Riem W A Q g where W denotes the Weyl curvature tensor and 0 is the natural extension of the exterior product to symmetric 0 2 -tensors usually referred to as the Kulkarni-Nomizu product Bes87 . Since the Weyl tensor is conformally invariant an important consequence of the decomposition is that the tran-formation of the Riemannian curvature tensor under conformal deformations of metric is completely determined by the transformation of the symmetric 0 2 -tensor A. In Via00a the second author initiated the study of the fully nonlinear equations arising from the transformation of A under conformal deformations. The research of the first author was partially supported by NSF Grant DMS-0200646. The research of the second author was partially supported by NSF Grant DMS-0202477. 476 MATTHEW J. GURSKY AND JEFF A. VIACLOVSKY More precisely let gu e 2ug denote a conformal metric and consider the equation 1-3 ơk k g-1 Au f x where ơk Rn R denotes the elementary symmetric polynomial of degree k Au denotes the Weyl-Schouten .