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We obtain a volume growth and curvature decay result for various classes of complete, noncompact Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual or Kahler metrics with zero scalar curvature, and metrics with harmonic curvature. Similar results are known for Einstein metrics, but our analysis differs from the Einstein case in that (1) we consider more generally a fourth order system in the metric, and (2) we do not assume any pointwise Ricci curvature bound.Comment: 54 pages; final versio

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.

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.

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