A Fully Nonlinear Equation on Four-Manifolds with Positive Scalar Curvature

Abstract

We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with a metric of positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. A special case of this deformation provides an alternative proof to the main result in [CGY02]. We also give a new conformally invariant condition for positivity of the Paneitz operator, generalizing the results in [Gur99]. From the existence results in [CY95], this allows us to give many new examples of manifolds admitting metrics with constant Q-curvature.

Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity

“…, n/2 if for instance (M, [h 0 ]) contains an Einstein metric in [h 0 ]; this is a result of Gover and Silhan [7]. If n = 4, L n/2−2 = L 0 is the Paneitz operator (up to a constant factor) and using a result of Gursky and Viaclovsky [14], we deduce that if the Yamabe invariant Y (M, [h 0 ]) is positive and…”

σk-Scalar curvature and eigenvalues of the Dirac operator

“…See [6] and [15]. Note that when n = 2m > 4 and the underlying manifold (M 2m , g) is locally conformally flat, it was proved in [22] that…”

On a fully nonlinear Yamabe problem

“…When n = 2k = 4, which is an important case, Chang-Gursky-Yang solved the problem in [7]. See also [6] and [23]. When the underlying manifold is locally conformally flat, this problem was solved by Guan-Wang [19] and Li-Li [29] independently.…”