An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature

By S. Chang, M. Gursky, Paul Yang. 2002

302 citations recorded by the source corpus.

We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively pinched.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/08333dc366741fdeb8d6c9fb02fd6f2937c050a7

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The Yamabe problem for higher order curvatures (2005, methodology): “As for the condition (C2), the variational approach to the k-Yamabe problem is natural, as in the case k = 1. Indeed this approach has already been employed in [CGY1, CGY2, GW2, GW3], and Theorem 2.1 was proved in [ CGY1 , CGY2] when n = 4, k = 2,”
Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds (2014, background): “= ψ(x)e (where Sg is the Schouten tensor of (M , g)) introduced by Viaclovsky [50] is connected with a natural fully nonlinear version of the Yamabe problem and has very interesting applications in conformal geometry; see for instance [5], [10], [26], [28], [35], [43] and references therein.”
Q‐Curvature on a Class of Manifolds with Dimension at Least 5 (2014, background): “7) Pe2wg' D e Pg': Following the basic work [2] in dimension 4 on the 4th orderQ curvature equation, there has been several studies on this equation in dimension 3 by [15, 36, 37], and in dimensions greater than 4 by [6, 21–23, 28, 29].”
Conformally invariant fully nonlinear elliptic equations and isolated singularities (2005, unknown): “See for instance [19], [4], [3], [12], [15], [10], and the references therein.”
Classification of singular radial solutions to the σk Yamabe equation on annular domains (2005, background): “[6] S.-Y. Chang, M. Gursky and P. Yang, Entire solutions of a fully nonlinear equation, “Lectures in Partial Differential Equations in honor of Louis Nirenberg’s 75th birthday”, chapter 3.”
On some conformally invariant fully nonlinear equations, II. Liouville, Harnack and Yamabe (2005, methodology): “Fully nonlinear equations involving the Schouten tensor have been studied by Viaclovsky in [71] and [70], and by Chang, Gursky and Yang in the remarkable papers [11] and [ 10 ].”
Compactness of solutions to some geometric fourth-order equations (2004, result): “To mention some geometric applications we recall three results proven by Gursky, [31], and by Chang, Gursky and Yang, [13], [14] (see also [30]).”
Recent developments in elliptic partial differential equations of Monge-Ampere type (2006, background): “The following theorem, due to Sheng, Trudinger and Wang [26] (k ≤ n/2), Gursky and Viaclovsky [16] (k > n/2) concerns the solvability of the higher order Yamabe problem, k > 1.”
A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound (2012, methodology): “involving the Schouten tensor and applications to geometry and topology have been studied extensively in and after the pioneering works of Viaclovsky [42, 43, 44] and Chang, Gursky and Yang [8, 9, 10, 11].”
Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂn (2004, background): “[15] Chang, S.-Y. A.; Yang, P. C.”
Singular sets of a class of locally conformally flat manifolds (2005, unknown): “For instance, Chang-Gursky-Yang [4], [3], Gursky-Viaclovsky [12], Li-Li [14], Guan-Wang [11].”