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International Mathematics Research Notices (2004) 2004:4269-4292, doi:10.1155/S1073792804141743
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Copyright © 2004 Hindawi Publishing Corporation. All rights reserved.

Local pointwise estimates for solutions of the {sigma}2 curvature equation on 4-manifolds

Zheng-Chao Han

The study of the kth elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called {sigma}k curvature, has produced many fruitful results in conformal geometry in recent years, especially when the dimension of the underlying manifold is 3 or 4. In these studies in conformal geometry, the deforming conformal factor is considered to be a solution of a fully nonlinear elliptic PDE. Important advances have been made in recent years in the understanding of the analytic behavior of solutions of the PDE, including the adaptation of Bernstein-type estimates in integral form, global and local derivative estimates, classification of entire solutions, and analysis of blowing-up solutions. Most of these results require derivative bounds on the {sigma}k curvature. The derivative estimates also require an a priori L{infty} bound on the solution. This work provides local L{infty} and Harnack estimates for solutions of the {sigma}2 bounds on the {sigma}2 curvature, and the natural assumption of small volume (or total {sigma}2 curvature).


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