Journal of Differential Geometry

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

Matthew J. Gursky and Jeff A. Viaclovsky


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 Chang, Gursky & Yang, 2002. We also give a new conformally invariant condition for positivity of the Paneitz operator, generalizing the results in Gursky, 1999. From the existence results in Chang & Yang, 1995, this allows us to give many new examples of manifolds admitting metrics with constant Q-curvature.

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J. Differential Geom., Volume 63, Number 1 (2003), 131-154.

First available in Project Euclid: 1 April 2004

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Gursky, Matthew J.; Viaclovsky, Jeff A. A Fully Nonlinear Equation on Four-Manifolds with Positive Scalar Curvature. J. Differential Geom. 63 (2003), no. 1, 131--154. doi:10.4310/jdg/1080835660.

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