Local pointwise estimates for solutions of the $\sigma_2$ curvature equation on 4 manifolds

Abstract

The study of the $k$-th 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, 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 solutoins. 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 $\sigma_2$ curvature equation on 4 manifolds, under only $L^p$ bounds on the $\sigma_2$ curvature, and the natural assumption of small volume(or total $\sigma_2$ curvature).